cyclobutadiene: what alternation costs and gains

One of the figures on hückel systems: Adjacency matrices diagonalised: levels, coefficients, bond orders, and the shell closures that decide which rings are stable.

Seven essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

The vibration that lowers the symmetry

Cyclobutadiene’s π energy as its bonds are alternated by β(1 ± δ), the elastic cost of alternating them at a stated stiffness, and the sum. The π curve falls linearly in the distortion — measured by halving the step, which leaves a linear difference quotient unchanged — so however stiff the frame, a small enough distortion wins.

Benzene, run through exactly the same code. Its π energy also falls when the bonds alternate — but quadratically, not linearly, because its shell is closed and no electron pair is waiting to be split apart. A stiff enough frame therefore holds it regular, and the figure at this stiffness does: the sum is highest at δ = 0.

The same ratio for every system in the calculation, which is the finding stated as a table rather than read off two curves. Every one of them gains π energy by alternating — that part is not the discriminator, and reading it as one is the mistake this figure exists to prevent. What separates them is the power: an open shell gains at first order, so there is no distortion small enough for a quadratic elastic cost to beat, and a closed shell gains at second order, so whether the flat ring survives is a competition that can go either way.

The hexagon is the frame's doing

Benzene’s π energy against bond alternation, with a σ restoring term subtracted. The π curve alone falls away from the symmetric geometry; whether the total has a minimum at zero depends entirely on how stiff the frame is, which is what the rest of this essay is about.

The gain and its exponent for four systems. Every gain is positive; the exponents sort them into the ones whose flat geometry is a stationary point and the ones whose is not, and no exponent was assumed — each is the slope of the log of the gain against the log of the distortion.

The critical force constant for benzene, over the decay lengths in use. It comes out between 6 and 12 millidyne per ångström, and the measured C–C stretching constant of benzene is about 7.6. The two are the same size, which is the honest end of this argument.

The distortion the filling chooses

The two sides of the competition against the size of the distortion: what the electrons gain and what the springs lose. The gain is the steeper of the two at small distortion and the loser at large, and where they cross is where the structure settles.

A distortion needs two states

And the molecular version: cyclobutadiene’s π energy against its bond alternation, falling linearly because its half-filled shell is degenerate. Every case in this essay has that term equal to zero and distorts anyway.

The frame that was allowed to relax

The one system where the relaxation and the symmetry argument disagree about the starting point: cyclobutadiene, whose undistorted form has a half-filled degenerate pair and is unstable against a distortion no amount of feedback on equal bond orders would find. A fixed point is not a minimum, and this is the shape of system where the difference shows.

An anomaly that is not the first of a series

Where the feedback between geometry and bond order starts, in the molecule where it is a distortion rather than a correction. The same arithmetic, run where the effect is first order instead of second.

The chain distorts hardest where it stops

The instability itself: the gain that beats any quadratic cost. Everything since has been about what the chain that gives way actually looks like.