Part 2 · a few hundred fs
Isomerization: Falling Through a Conical Intersection
Absorption puts retinal on its first excited electronic surface, S1, with the all-trans geometry of the ground state. On S1 the C13=C14 bond loses much of its double-bond character, so twisting around it costs little energy and the molecule starts to rotate.
Potential energy surfaces
In the Born–Oppenheimer picture, each electronic state defines an energy surface over the nuclear coordinates, here mainly the torsion angle θ of the C13=C14 bond. The ground state S0 has minima at θ = 0° (trans) and 180° (cis) with a high barrier at 90°; the excited state S1 slopes downhill toward 90°.
The conical intersection
Near θ ≈ 90°, S1 and S0 become degenerate along a seam where the two surfaces touch like the tips of two cones. There the Born–Oppenheimer approximation breaks down: nuclear motion couples the electronic states strongly, and the wavepacket crosses from S1 to S0 in a single vibrational period. Emerging on the ground state at the top of the barrier, the molecule rolls either back to trans or forward to 13-cis.
The ideas behind the picture
The Born–Oppenheimer approximation
A carbon nucleus is about 1836 × 12 ≈ 22,000 times heavier than an electron. For the same kinetic energy, its speed is smaller by the square root of that ratio, about 150 times. Electrons therefore rearrange almost instantly whenever the nuclei move. We can freeze the nuclei at positions R, solve for the electronic energy E(R), and treat that energy as the potential in which the nuclei move. Plotting E(R) for each electronic state gives the surfaces in Figure 2.
The Franck–Condon principle
Absorbing a photon takes about one optical period, roughly 2 fs for blue light. In that time the nuclei barely move, so the transition is vertical on the diagram: the molecule lands on S1 with the ground-state geometry, high up a slope. The excess energy is what drives the twist.
Three clocks, three timescales
| Motion | How to estimate it | Period |
|---|---|---|
| Electronic, absorption | h / ΔE with ΔE ≈ 2.6 eV | ≈ 1.6 fs |
| C=C bond stretch | 1 / (c ν̃) with ν̃ ≈ 1550 cm⁻¹ | ≈ 21 fs |
| Twist about C13=C14 | heavy groups rotating through 90° | ≈ 100–500 fs |
The twist is the slowest step, so it sets the speed of the whole primary reaction. Yet it is still about a million times faster than the millisecond opening of the channel.
Why fast matters
An excited molecule that lingers can lose its energy in unproductive ways: emitting fluorescence (nanoseconds) or heating its surroundings (picoseconds). By reaching the funnel within a few hundred femtoseconds, retinal beats these losses, which is why most absorbed photons do useful work.
Speed and quantum yield
The fraction of absorbed photons that end in the cis form is the quantum yield; in visual rhodopsin it is about 0.65 (an approximate literature value, used here as a teaching estimate). The protein pocket steers the wavepacket so that the forward branch is favoured, and its rigid walls store part of the photon's energy as strain in the twisted chromophore.
Derivation: why the surfaces form a cone
Step 1 — Two-state Hamiltonian
Near the crossing, keep only the two electronic states that matter. In a basis of two diabatic states, with energies E₁(R) and E₂(R) and coupling V(R) that all depend on the nuclear coordinates R:
\[ H(\mathbf{R}) = \begin{pmatrix} E_1(\mathbf{R}) & V(\mathbf{R}) \\ V(\mathbf{R}) & E_2(\mathbf{R}) \end{pmatrix} \]
Step 2 — Diagonalise
The adiabatic surfaces S₀ and S₁ are the eigenvalues:
\[ E_\pm(\mathbf{R}) = \frac{E_1 + E_2}{2} \pm \sqrt{\left(\frac{E_1 - E_2}{2}\right)^2 + V^2} \]
Step 3 — Count the conditions
The two surfaces touch only if the square root vanishes, which needs two independent conditions at once: E₁ = E₂ and V = 0. In a molecule with M internal coordinates, the degeneracy therefore survives on a seam of dimension M − 2. For a diatomic, M = 1, so the surfaces cannot cross: the non-crossing rule.
Step 4 — The double cone
Expand to first order around a point on the seam, along the two directions that lift the degeneracy: x along the gradient difference g and y along the coupling gradient h:
\[ E_\pm(x, y) \approx E_0 + \mathbf{s}\cdot\mathbf{r} \pm \sqrt{(g\,x)^2 + (h\,y)^2} \]
The ± square-root term is the equation of two cones meeting at their tips: hence the name.
Step 5 — Born–Oppenheimer breaks down
The coupling that drives jumps between surfaces is the non-adiabatic coupling vector. Using the Hellmann–Feynman theorem:
\[ \mathbf{d}_{01} = \langle \psi_0 \,|\, \nabla_{\mathbf{R}}\, \psi_1 \rangle = \frac{\langle \psi_0 \,|\, \nabla_{\mathbf{R}} H \,|\, \psi_1 \rangle}{E_1 - E_0} \]
As the gap E₁ − E₀ goes to zero, this coupling diverges: electrons can no longer follow the nuclei adiabatically, and the wavepacket crosses to S₀.
Step 6 — Hopping probability
In the Landau–Zener model, a system passing a crossing at speed v hops to the other adiabatic surface with probability:
\[ P_{\text{hop}} = \exp\!\left(-\frac{2\pi V^2}{\hbar\, v\, |\Delta F|}\right) \]
where ΔF is the difference in slopes of the diabatic curves. At a conical intersection V → 0, so P → 1: a fast wavepacket goes straight through the funnel. This is why retinal relaxes to the ground state in a single pass, within a few hundred femtoseconds.
Key idea for students
A conical intersection is to photochemistry what a transition state is to thermal chemistry: the funnel that sets the rate and the outcome. Here a protein has evolved to shape that funnel.
What this model leaves out
A single torsion coordinate stands in for a molecule with dozens of internal degrees of freedom, and the surfaces in Figure 2 are schematic shapes, not computed ones. The Landau–Zener formula also assumes a linear sweep through the crossing at constant speed, which a real wavepacket only approximates.