= Solution
The <Hückel orbital charge density> and <Hückel pi bond order> give
$$
n_i=\sum_k f_kc_{ik}^2,\qquad p_{ij}=\sum_k f_kc_{ik}c_{jk}.
$$
Each site receives $1/2$ <electron> from the lowest orbital and one from the occupied nonbonding subspace, so \b[$n_i=3/2$ at every <carbon>], summing to six. Relative to one neutral pi <electron> per <carbon> this is a pi charge of $-e/2$ at each site.
For adjacent sites, only the lowest orbital contributes, giving \b[$p_{12}=p_{23}=p_{34}=p_{41}=1/2$]. For opposite sites the nonbonding contribution reverses the sign: \b[$p_{13}=p_{24}=-1/2$]. These non-neighbour negative density-matrix entries are not extra <sigma bonds> or extra resonance integrals in the <Hamiltonian>. If only skeletal nearest-neighbour <pi bond> orders are requested, the four values $1/2$ are the relevant ones.
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