advanced28 min studyLesson 468 in Physics

Conical intersections and nonadiabatic coupling

Resolve a two-surface degeneracy in its branching plane and distinguish adiabatic energies, diabatic couplings, and actual nuclear transition dynamics.

Before you begin

On this page

The question for today

What you’ll understand

Diagonalize a local real two-state conical-intersection model.
Explain why two independent coordinates lift a generic degeneracy.
Identify the representation dependence of couplings and the limitations of a surface diagram.

Some molecular geometries bring two electronic surfaces to the same energy. Near such a degeneracy, nuclear motion and electronic state character can become strongly entangled, undermining an independent single-surface picture. A conical intersection is especially important because moving in two independent nuclear directions generally lifts the degeneracy linearly. We will diagonalize a local two-state model, explain the difference between an avoided crossing on one chosen path and a true intersection in a larger coordinate space, and avoid inferring a reaction probability from geometry alone.

Two conditions define a real two-state degeneracy

For a real symmetric two-state electronic Hamiltonian, degeneracy requires its diagonal entries to match and its off-diagonal entry to vanish. These are two independent conditions in the generic spin-free setting considered here. A molecule with several internal nuclear coordinates can satisfy both along a seam; two branching directions move it away from the seam. A diatomic has only one internal separation coordinate, so same-symmetry surfaces generically avoid each other along that coordinate unless additional symmetry or special structure changes the conditions.

Choose local branching coordinates x and y and neglect a common tilt for clarity. One coordinate changes the difference of diagonal energies and the other changes the electronic coupling. Diagonalization gives a square-root splitting proportional to distance in a rescaled branching plane. This is the double-cone shape behind the name. It is a local linear model, not a global molecular surface, and spin-orbit coupling, complex Hamiltonians, or symmetry restrictions can change the degeneracy classification.

H_{\mathrm{loc}}=E_0I+\begin{pmatrix}\kappa x&\lambda y\\\lambda y&-\kappa x\end{pmatrix},\qquad E_\pm=E_0\pm\sqrt{\kappa^2x^2+\lambda^2y^2}

x and y are nuclear-coordinate displacements with stated length units, and kappa and lambda are energy per length. At x = y = 0 the real spin-free two-state model is degenerate; it omits a common tilt and higher-order geometry dependence.↔ Scroll wide equations horizontally.

Adiabatic and diabatic representations organize different terms

Adiabatic electronic states diagonalize the clamped Hamiltonian at each geometry. Their eigenvectors change rapidly near the intersection, giving large nuclear derivative couplings. Away from degeneracy, differentiating the electronic eigenproblem expresses an off-diagonal derivative coupling through a Hamiltonian derivative divided by the electronic energy gap. That denominator explains why a mass-based separation can fail in a narrow region even when the surrounding surfaces behave smoothly.

A locally smooth diabatic-like basis keeps state character more continuous and places an explicit off-diagonal potential coupling in the electronic matrix. The two representations distribute terms differently while describing the same full coupled problem when treated consistently. Constructing globally exact diabatic states is not guaranteed. Singular derivative couplings in a particular adiabatic representation do not imply an infinite measurable transition probability; the coupled nuclear wavefunction must be evolved with the appropriate terms and boundary conditions.

\mathbf d_{ab}=\langle\phi_a|\nabla_R\phi_b\rangle=\frac{\langle\phi_a|\nabla_RH_e|\phi_b\rangle}{\varepsilon_b-\varepsilon_a}\quad(a\ne b,\ \varepsilon_a\ne\varepsilon_b)

The identity is valid for differentiable nondegenerate electronic eigenstates with a consistent local phase convention. It cannot be evaluated by setting the denominator to zero at the intersection; the degenerate subspace then needs joint treatment.↔ Scroll wide equations horizontally.

A surface picture is not a branching probability

Taking a one-dimensional path with y fixed and nonzero produces an avoided crossing with a finite minimum gap. Taking a path through the origin can reach the true degeneracy. Both are cuts through the same two-dimensional model, so apparently contradictory plots can describe different nuclear paths. Nuclear momentum, wavepacket width, approach direction, coherence, and energy all affect population transfer. A local gap value or a cone sketch alone cannot supply a unique probability of ending on a particular surface.

In the ideal real two-state conical model, transporting an adiabatic eigenvector continuously around the intersection can change its sign, corresponding to a geometric phase of pi. A global sign of a lone state is unobservable, but relative phases of nuclear paths can affect interference when branches reunite. This topology motivates coupled and phase-consistent nuclear treatments. It does not mean every photochemical pathway passes through a conical intersection or that an intersection itself determines a unique chemical product.

Work it through

A point beside the intersection

In the local model let $\kappa x=0.030\,\mathrm{eV}$ and $\lambda y=0.040\,\mathrm{eV}$ . Find both energies relative to E zero and their gap. Then hold y fixed and move x through zero to find the minimum gap on that path.

The square-root splitting is $\sqrt{0.030^2+0.040^2}=0.050\,\mathrm{eV}$ , using the three-four-five ratio.
The adiabatic energies are therefore $E_0-0.050\,\mathrm{eV}$ and $E_0+0.050\,\mathrm{eV}$ , giving total gap 0.100 eV. The square root is half the gap, not the whole gap.
With y fixed, the minimum occurs at x = 0 and the gap is $2|\lambda y|=0.080\,\mathrm{eV}$ . This chosen one-dimensional cut avoids the degeneracy even though the full two-coordinate surface intersects at the origin.

The result

The selected point has a 0.100 eV gap; the fixed-y path has minimum gap 0.080 eV. Neither number alone determines a nuclear population-transfer probability.

The living notebook

Explore the model

Interactive lab

Interactive lab · quantum

Electronic phase is part of a coupled description

Reset, set Mixing angle θ = 90° and Relative phase φ = 0°, and inspect the initial Bloch direction. For a fresh comparison Reset, set θ = 90° and φ = 90°, and predict unchanged equal basis probabilities but a rotated Bloch vector. Inspect before playback adds further phase.

Bloch vector / P(0)P(1)Phase latitude

Probability |0⟩: 71.1%
Probability |1⟩: 28.9%
Bloch (x, y, z): 0.78, 0.45, 0.42

Mixing angle θ65°Relative phase φ30°

t = 0 s

A normalized two-level pure state, not an interacting field simulation. Playback adds one radian per second to the relative phase; basis probabilities stay fixed. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThe two-state lab illustrates relative phase only. It has no nuclear coordinates, conical surfaces, derivative couplings, Berry loop, or Landau–Zener transition solver; changing controls does not move a molecule through a crossing.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

A fixed nonzero coupling coordinate keeps a finite gap along a one-dimensional cut. The full branching plane contains a point where diagonal difference and off-diagonal coupling both vanish.

Your turn

Try it yourself

Use $\kappa=2\,\mathrm{eV\,\mathring A^{-1}}$ and $\lambda=1\,\mathrm{eV\,\mathring A^{-1}}$ . At $x=0.030\,\mathring A$ and $y=0.080\,\mathring A$ , find the gap. Then find the gap at x = 0 with $y=0.020\,\mathring A$ .

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Convert each coordinate to its energy contribution first; the eigenvalue gap is twice the square root.

Compare with the solution

The first contributions are 0.060 and 0.080 eV, so the half-gap is 0.100 eV and the gap is 0.200 eV. At the second point the diagonal difference term vanishes and the half-gap is 0.020 eV, giving gap 0.040 eV. Only setting both x and y to zero reaches the model’s degeneracy.

Keep the idea

Take it with you

A generic real two-state intersection requires two independent conditions and has a branching plane.
Adiabatic derivative couplings and diabatic potential couplings are representation-dependent parts of one coupled problem.
Energy surfaces guide dynamics, but gaps and geometry alone do not determine chemical branching probabilities.

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