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Sandbox Physics
Nonadiabatic molecular dynamics

L3 · Chapter 5

How two surfaces form a conical intersection

Sandbox Physics ·

Two electronic energy surfaces can meet exactly at a nuclear configuration. One direction reduces their energy difference; an independent direction changes their mixing. Plot both directions and the contact looks like two opposing cones.

A generic single coordinate cannot satisfy both conditions

A real symmetric two-state matrix has common energy ss, half-difference zz and coupling cc. Degeneracy requires z=0z=0 and c=0c=0. A generic one-coordinate scan cannot independently tune both, so a finite coupling leaves an avoided crossing.

Introduce a tuning coordinate xx and mixing coordinate yy. Locally set z=axz=ax and c=byc=by with nonzero energy gradients aa and bb. The degeneracy lies at the origin.

V(x,y)=s(x,y)I+(axbyby−ax),E±=s±a2x2+b2y2V(x,y)=s(x,y)I+\begin{pmatrix}ax&by\\by&-ax\end{pmatrix},\qquad E_\pm=s\pm\sqrt{a^2x^2+b^2y^2}
This is a local real two-state model without spin–orbit coupling. Complex couplings, additional states and symmetry can change the degeneracy structure.
ΔE=2a2x2+b2y2\Delta E=2\sqrt{a^2x^2+b^2y^2}
The gap grows linearly with distance from contact along either independent direction.

Both coordinates matter to a passing packet

Head-on and off-center packets encounter different gaps and changes of electronic character. A packet has transverse width, so even a center aimed at the degeneracy carries amplitude past both sides. A single central trajectory loses that structure.

The existing conical-intersection experiment evolves a two-component wavefunction in a dimensionless linear vibronic-coupling model with common potential s=ω2(x2+y2)/2s=\omega^2(x^2+y^2)/2. The fixed dimensionless confinement frequency is 0.65; it controls the strength of the common harmonic potential. Surface color and readouts must refer consistently to either fixed channels or adiabatic projections.

Worked example of an offset straight path

Prescribe x=vtx=vt and y=b0y=b_0 with v>0v>0. The half-difference is avtavt and coupling is bb0bb_0. The minimum gap is 2∣bb0∣2|bb_0|, so doubling the offset doubles the gap.

Under the additional assumptions of one uniform, asymptotically long passage, the Landau–Zener parameter is Γ=b2b02/(2ℏ∣av∣)\Gamma=b^2b_0^2/(2\hbar|av|). The full diabatic difference changes at rate 2av2av. Doubling offset therefore quadruples the exponent’s magnitude.

This estimates a prescribed local path. A two-dimensional nuclear packet spreads and bends, with different components sampling different couplings. Full propagation is needed for its branching probabilities.

Reading: C. Zener

Electronic population depends on the projector

A diabatic channel probability integrates the squared magnitude of a fixed component. Lower-adiabatic population first applies the local projector P−(x,y)P_-(x,y). They answer different questions and may have different values.

Total density adds both component densities and is preserved by a consistent unitary change of basis. At the degeneracy the individual eigenstate is not unique. Its zero-measure contribution vanishes in continuous integrals; a finite grid needs a separately checked convention.

P−(t)=∫dx dy  χ†(x,y,t)P−(x,y)χ(x,y,t)P_-(t)=\int d x\,d y\;\boldsymbol\chi^\dagger(x,y,t)P_-(x,y)\boldsymbol\chi(x,y,t)
The left side denotes integrated lower-surface population; the matrix inside the integral is the local projector.

After branching, ask what happens to phase

Separate density lobes show nuclear branching. If amplitudes meet again, their relative phase also controls the density. Paths around a conical intersection can recombine destructively.

To identify geometric phase, compare electronic states around a closed circuit or controlled interference with the phase retained and removed. A change in one instantaneous surface population does not isolate that effect. The next chapter specifies paths and recombination so the phase can be tested separately.

Reading: M. V. Berry

Try the prediction

Change the incident offset

  1. Predict which preparation samples the smallest gaps: head-on or off-center.
  2. Change only the offset and compare density and lower-surface population at the same time.
  3. Use the no-mixing control and explain which electronic transfer disappears while nuclear evolution continues.

Exercises and explanations

  1. Why does a generic real two-state scan with one coordinate avoid crossing?

    Read the explanation

    Degeneracy requires both the diagonal difference and real off-diagonal coupling to vanish. These are generally two independent conditions.

  2. Do different diabatic and adiabatic populations imply a failure of probability conservation?

    Read the explanation

    They use different projectors. Check the sum over each complete set against total probability; individual channels should be projected onto the same electronic states before comparison.

Sources and calculation scope

  1. M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A 392, 45–57 (1984)

    Published primary derivation: equations (5)–(6) separate dynamical and geometrical phase; section 3 treats a degeneracy. The path experiment is a controlled two-state model, with the reduction and phase comparison specified on its page.

  2. C. Zener, Non-adiabatic crossing of energy levels, Proc. R. Soc. Lond. A 137, 696–702 (1932)

    Published original result for a linear-in-time diabatic gap and constant coupling. The course defines the slope as the slope of the full diabatic energy difference, avoiding a factor-of-two ambiguity.

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