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Sandbox Physics

Two energy surfaces / One wave packet

Through a Conical Intersection

What happens to a molecular wavepacket when it passes the point where two energy surfaces touch? It splits between the two surfaces. Near the touching point the gap between them closes, so the electronic state cannot smoothly follow the moving nuclei; part of the packet drops onto the lower surface and part stays on the upper one. How it splits, and the shape each part takes, depend on how close to the point it passes, how fast it moves and how strongly the two electronic states mix.

Send a nuclear wave packet toward a conical intersection between two electronic states. Compare a head-on arrival, an off-centre pass and a control run with no mixing, using the population on each surface and independently sampled configurations.

Interactive modelThrough a Conical Intersection
Electronic energy across nuclear configurations

LOWER ENERGY SURFACE POPULATION

0%0\%

EVOLUTION TIME

τ=0\tau=0

Read the probability

The surface is calculated; the dots are simulated measurements.

The nuclear configuration plane

Cyan is the lower energy surface and coral the upper. Brightness is the square root of the total density, on a fixed scale. The axes are nuclear coordinates, not electron paths.

Lower00.7
Upper00.7

Two bases, two different readings

Solid cyan is the population on the local lower surface; dashed violet is the population in fixed channel A. They are different projections. Gold is the lower-surface population at the pinned instant.

ENERGY DRIFT00
NORM DRIFT00
EDGE WEIGHT00
HIGH-FREQUENCY WEIGHT00

Independent simulated measurements

Pick an instant, then measure freshly prepared molecules

Sample nuclear configuration and fixed electronic channel from each of 768 independent preparations. Violet dots are channel A; gold dots are B. These are fixed electronic channels, not energy surfaces.

Ideal joint position readout, without blur or loss. Moving the time cursor preserves old records; changing preparation clears them. Samples are not a wavefunction reconstruction or a continuous movie of one molecule.

Equations, coordinates and boundaries
x=q/ℓ,τ=ℏtmℓ2,E∗=ℏ2mℓ2x=q/\ell,\quad\tau=\frac{\hbar t}{m\ell^2},\quad E_* =\frac{\hbar^2}{m\ell^2}

Length, time and energy are dimensionless. No specific molecular mass or experimental time scale is assigned. Positive stretch points outward; in the two-coordinate model the second coordinate is positive upward in the flat map.

H=−12∇2I+V0I+xσz+κyσxV0=12ω2(x2+y2),ω=0.65V±=V0±x2+κ2y2\begin{gathered}H=-\tfrac12\nabla^2 I+V_0 I+x\sigma_z+\kappa y\sigma_x\\V_0=\tfrac12\omega^2(x^2+y^2),\quad\omega=0.65\\V_\pm=V_0\pm\sqrt{x^2+\kappa^2y^2}\end{gathered}

The full two-component state is propagated in a fixed diabatic basis. Lower-surface density uses the local spectral projector; at an exactly degenerate grid point its weight is assigned equally. Setting mixing to zero removes electronic-channel transfer and changes the point intersection to a line crossing. No claim of measuring a Berry phase is made.

Ψ(x,y,0)=(0g(x,y))g∝e−[(x+4)2+(y−y0)2]/(2s2)eip0x,s=0.7\begin{gathered}\Psi(x,y,0)=\begin{pmatrix}0\\g(x,y)\end{pmatrix}\\g\propto e^{-[(x+4)^2+(y-y_0)^2]/(2s^2)}e^{ip_0x},\quad s=0.7\end{gathered}

A periodic Fourier grid and symmetric split-step propagation solve the declared Hamiltonian. No absorption or propagation renormalization is used. Edge occupation, high-wave-number weight and relative energy drift are checked; the view warns if the chosen run loses numerical resolution.

Sources establish the model context; this illustrative parameter set is not a fit or a reproduction of those experimental data.

Physics tutorial

Through a Conical Intersection

BackgroundSend a nuclear wave packet toward a conical intersection between two electronic states. Compare a head-on arrival, an off-centre pass and a control run with no mixing, using the population on each surface and independently sampled configurations.

Why it mattersLink the shape of the energy landscape to the way the probability moves, then ask what a measurement would actually record.

Start with the essentials

Focus question
What happens to a molecular wavepacket when it passes the point where two energy surfaces touch?
One-sentence intuition
The electronic mixing and the nuclear motion change together. Staying in a fixed electronic channel is not the same as staying in the locally lower or upper energy state.

Core mathematical model

Hamiltonian

H=−12∇2I+ω2(x2+y2)2I+xσz+κyσxH=-\frac12\nabla^2 I+\frac{\omega^2(x^2+y^2)}2I+x\sigma_z+\kappa y\sigma_x

In dimensionless units the effective mass and the reduced Planck constant are both one. The time evolution is unitary, and the wavefunction is never renormalized between steps.

Energy surfaces

V±=ω2(x2+y2)2±x2+κ2y2V_\pm=\frac{\omega^2(x^2+y^2)}2\pm\sqrt{x^2+\kappa^2y^2}

With mixing switched on, the two surfaces touch at a single point. With mixing set to zero they cross along a line instead, so the control changes the geometry of the model as well as its dynamics.

Common difficulties

A landscape is not a trajectory

Typical misconceptionThe bright spot is a single atom moving across a real surface.

Better mental modelIt shows the probability of each nuclear configuration. Drawing energy as height is only a way to visualize it.

What the model covers

Typical misconceptionThe simulation predicts a particular molecule.

Better mental modelThe model is a dimensionless linear vibronic-coupling model in two coordinates, evolved as a two-component wavefunction. The axes are nuclear configuration coordinates and the height is electronic potential energy. The picture is not an atom's trajectory, it does not model a particular molecular reaction, and it is not a direct measurement of geometric phase.

Run the experiment

  1. 01

    Compare

    Start with the first two presets and select the same time in both. Pin a frame before changing the preparation.

    What to observe: An off-centre start changes both the spatial pattern and the populations.
  2. 02

    Read the axes

    Switch between landscape and density views, then inspect the data row.

    What to observe: The height of the potential, the brightness of the density and the probability relief are different quantities, each with its own labelled scale.
  3. 03

    Take samples

    Sample independent preparations at one chosen instant. Move time and confirm the old records stay fixed. Export the raw CSV.

    What to observe: A finite sample fluctuates. It cannot rebuild the complex wavefunction, and it does not show a continuous path.

Related experiments and articles

Nonadiabatic molecular dynamics · Reading path

From energy surfaces to reaction branches

Articles on this path that use this experiment, followed by the other experiments they suggest.

  1. ArticleHow two surfaces form a conical intersection
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