 Solution

Use the printed numbering: site 1 is at the top on $+y$, sites 2 and 3 are on the right, site 4 is at the bottom, and sites 5 and 6 are on the left; $z$ is perpendicular to the ring. Let ${\chi }_j$ be consistently phased, orthonormal $p_z$ atomic orbitals at those sites.

In the Hückel method, $H_{jj}=\alpha$ and $H_{j,j\pm 1}=\beta$, including the 6–1 bond. Set $\lambda =(\epsilon -\alpha )/\beta$, with the usual $\beta <0$. The cyclic nearest-neighbour matrix has eigenvalues ${\lambda }_k=2\cos (2\pi k/6)$. One real, normalized orthonormal eigenbasis isEach row means $\psi ={\sum }_j{c}_j{\chi }_j$. The three lowest molecular orbitals hold the six $\pi$ electrons. Any real orthogonal mixing within either degenerate pair is an equally valid choice of molecular orbitals; the coefficient vectors specify exactly which sketches are used here.

The coloured circles show the signs and relative magnitudes of the lobes above the molecular plane. The lobes below it have the opposite signs; colours are quantum phases, not electric charges.