  For a symplectic space $V$ of dimension $2n$ over $\mathbb{F}_{q}$, we
compute the eigenvalues of its orthogonality graph. This is the simple graph
with vertices the $2$-dimensional non-degenerate subspaces of $V$ and edges
between orthogonal vertices. As a consequence of Garland's method, we obtain
vanishing results on the homology groups of the frame complex of $V$, which is
the clique complex of this graph. We conclude that if $n < q+3$ then the poset
of frames of size $\neq 0,n-1$, which is homotopy equivalent to the frame
complex, is Cohen-Macaulay over a field of characteristic $0$. However, we also
show that this poset is not Cohen-Macaulay if the dimension is big enough.
