TWO SURFACES / ONE WAVEPACKET

Through a Conical Intersection

Send a nuclear wavepacket toward a two-state conical intersection. Compare a central arrival, an off-centre passage and a zero-mixing control using surface populations and independently sampled configurations.

Dimensionless two-coordinate linear vibronic-coupling model, evolved as a two-component wavefunction. The axes are nuclear configuration coordinates; height is electronic potential energy. This is not an atom trajectory, a particular molecular reaction or a direct measurement of geometric phase.

NUCLEAR CONFIGURATION / ELECTRONIC ENERGY

LOWER ENERGY SURFACE POPULATION

0\%

EVOLUTION TIME

\tau=0

READ THE PROBABILITY

The surface is a model. The dots are samples.

The nuclear configuration plane

Cyan indicates the lower energy surface, coral the upper. The total brightness is square-root density with a fixed scale. These are nuclear coordinates, not electron tracks.

Lower00.7

Upper00.7

Two bases, two different readings

Solid cyan: local lower-surface population. Dashed violet: fixed channel A population. These are different projections. Gold is the pinned lower-surface population.

ENERGY DRIFT

0

NORM DRIFT

0

EDGE WEIGHT

0

HIGH-FREQUENCY WEIGHT

0

INDEPENDENT SIMULATED MEASUREMENTS

Stop time. Prepare again. Measure.

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/\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.

\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.

\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.

Start with the essentials

Focus question: Two energy landscapes touch. The wavepacket takes more than one way.
One-sentence intuition: Electronic mixing and nuclear motion evolve together. A fixed electronic channel is different from the locally lower or upper energy state.

Core mathematical model

Hamiltonian

H=-\frac12\nabla^2 I+\frac{\omega^2(x^2+y^2)}2I+x\sigma_z+\kappa y\sigma_x

Dimensionless units set the effective mass and reduced Planck constant to one. All propagation is unitary; no time-step renormalization is used.

Energy surfaces

V_\pm=\frac{\omega^2(x^2+y^2)}2\pm\sqrt{x^2+\kappa^2y^2}

With nonzero mixing the surfaces meet at one point. Setting mixing to zero produces a crossing line, so it changes the model geometry as well as the dynamics.

Common difficulties

A landscape is not a trajectory

Typical misconceptionThe bright spot is one atom moving on a literal surface.

Better mental modelIt is a nuclear configuration probability. Energy is plotted vertically only as a visualization.

Scope

Typical misconceptionThe simulation predicts a particular molecule.

Better mental modelDimensionless two-coordinate linear vibronic-coupling model, evolved as a two-component wavefunction. The axes are nuclear configuration coordinates; height is electronic potential energy. This is not an atom trajectory, a particular molecular reaction or a direct measurement of geometric phase.

Run the experiment

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 preparation changes both the spatial pattern and the populations.
02
Read the axes
Switch between landscape and density views, then inspect the data row.
What to observe: Potential height, density brightness and probability relief are distinct quantities with explicit display scales.
03
Acquire
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 does not reconstruct the complex wavefunction or reveal a continuous path.