Every site has one short-bond and one long-bond neighbour. Equivalently, apply the Hückel method matrix to any coefficient and read off its two neighbouring signs. The results are

symmetry A_{1 u} A_{2 u} B_{1 u} B_{2 u} E α - β_{S} - β_{L} α + β_{S} + β_{L} α - β_{S} + β_{L} α + β_{S} - β_{L}

(3)

With negative resonance integrals and

∣ β_{S} ∣ > ∣ β_{L} ∣

, the

A_{2 u}

and

B_{2 u}

levels are occupied, while

B_{1 u}

and

A_{1 u}

are unoccupied in the closed-shell alternating-bond state.

Figure 1.
Original phase patterns for the four one-dimensional carbon-ring symmetry orbitals in the printed numbering and axes
. **Details**.

Solution

 0  0

Use the four supplied opposite-site combinations in the order

(ϕ_{1}, ϕ_{2}, ϕ_{3}, ϕ_{4})

. Their Hückel method matrix is

H_{E_{g}} = α β_{S} 0 - β_{L} β_{S} α β_{L} 0 0 β_{L} α β_{S} - β_{L} 0 β_{S} α .

(1)

The negative corner is essential. For example,

⟨ ϕ_{4} ∣ H ∣ ϕ_{1} ⟩

includes the long bonds

4

–

5

and

8

–

1

, each with opposite coefficient signs, giving

- β_{L}

Because the basis is orthonormal, the secular determinant is

α - E β_{S} 0 - β_{L} β_{S} α - E β_{L} 0 0 β_{L} α - E β_{S} - β_{L} 0 β_{S} α - E = 0.

(2)

It factors as

[(α - E)^{2} - β_{S}^{2} - β_{L}^{2}]^{2} = 0

, giving two doubly degenerate energies,

α \pm β_{S}^{2} + β_{L}^{2}

The four coefficient vectors are mutually orthonormal. Any orthogonal mixing within either equal-energy pair is an equally valid choice of molecular orbitals. At this limiting parameter value, additional coincidences with the one-dimensional levels occur: the full eight-orbital system is four isolated dimers, with four bonding and four antibonding levels.