A molecular wavepacket chooses its exits

Three Ways Through an Avoided Crossing

Do quantum waves, a mean force and hopping trajectories predict the same reaction branches? A nuclear wavepacket can split between electronic states near an avoided crossing. Quantum evolution keeps both amplitudes; Ehrenfest follows their average force, and surface hopping samples separate paths. Their outgoing probabilities can differ.

Send the same nuclear preparation through two coupled electronic states. Watch the quantum packet split, compare the mean-force swarm and count trajectories that switch surfaces.

Model assumptions

Tully’s one-coordinate, two-state model uses atomic units and an effective nuclear mass of 2000 electron masses. It is not calibrated to a molecule. The quantum packet and classical Wigner ensemble share the same initial Gaussian; their finite-time exit probabilities are compared at the same windows. Surface hopping retains electronic coherence and adds no decoherence correction.

t=0

One nuclear coordinate. Height shows electronic energy; ribbon thickness shows density on a separate scale. Cyan is lower, coral is upper.

At the common final time

Follow the probability into the exits

The exit windows begin four bohr from the crossing. Channel estimates include probability absorbed far beyond its exit; surviving and absorbed parts are saved separately. Probability between the windows is unresolved.

Quantum upper exit

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Hopping minus quantum

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Hopping standard error

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Quantum unresolved

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Four exit probabilities

Bars share the same probability scale. Thin whiskers show one sampling standard error for the trajectory estimates.

Population reaching the right exit

Quantum, mean-force and hopping predictions use the same two exit windows. An incoming packet on the left is not yet a reflected reaction product.

Norm, energy, flux and measured cost

Probability still on the grid

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Probability absorbed outside the exits

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Norm plus absorbed

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Quantum energy drift

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Largest trajectory energy drift

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Net outward window flux

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Quantum / mean force / hopping milliseconds

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Net outward flux subtracts incoming flux through the left window. Initially the packet is outside the windows: final exterior probability plus absorbed loss equals initial exterior probability plus net outward flux. Trigonometric interpolation evaluates current at the exact window planes; fractional-cell quadrature integrates their densities.

Costs measure propagation on this device, excluding snapshots, figures and downloads. Quantum energy drift is meaningful before absorption; the absorbing layer removes probability beyond nineteen bohr.

Model and numerical limits

The quantum solver uses 512 periodic Fourier points on a forty-eight-bohr box and a Strang time step no larger than 0.8 atomic time units. It applies the exact local two-state exponential and an outer absorbing mask. No stepwise normalization is applied. The offline benchmark halves the step and doubles the grid.

Ehrenfest trajectories follow the electronic mean force. Fewest-switches trajectories follow one adiabatic surface, propagate electronic amplitudes coherently, rescale momentum at accepted hops and reject energetically forbidden hops without velocity reversal. No decoherence correction is added.

Read the benchmark method

Start with the essentials

Focus question: Do quantum waves, a mean force and hopping trajectories predict the same reaction branches?
One-sentence intuition: A mean-force trajectory feels both electronic branches; a hopping trajectory moves on one surface at a time. The quantum packet carries both amplitudes.

Core mathematical model

A nuclear coordinate coupled to two electronic states

H=-\frac{1}{2M}\partial_x^2 I+\begin{pmatrix}z(x)&w(x)\\w(x)&-z(x)\end{pmatrix},\quad M=2000

Atomic units set the reduced Planck constant to one. Position is in bohr, energy in hartree and time in atomic time units. The matrix eigenvalues are the two adiabatic energies.

The original simple avoided crossing

z(x)=0.01\,\operatorname{sgn}(x)(1-e^{-1.6|x|}),\qquad w(x)=0.005c\,e^{-x^2}

Electronic mixing multiplies the original off-diagonal coupling. The original Tully model has mixing equal to one. The initial packet is on the lower adiabatic surface.

A shared Gaussian preparation

\chi(x)\propto e^{-(x+8)^2/(4\sigma_x^2)+ip_0x},\quad \sigma_p=\frac{1}{2\sigma_x}

The classical methods independently sample the positive Wigner distribution of this same minimum-uncertainty packet. Their initial electronic state is the local lower eigenvector.

Common difficulties

A population is not an exit probability

Typical misconceptionElectronic population alone gives the reaction yield.

Better mental modelAn exit also requires the nuclear packet to reach its chosen spatial window. During the animation the incoming packet is still on the left; the table uses the common final time and reports unresolved weight separately.

Run the experiment

01
Watch the split
Choose Two outgoing branches and play the evolution. Compare the three rows at the same time.
What to observe: The quantum density spreads onto both energy curves. Surface-hopping trajectories separate by their active surface; mean-force trajectories can travel between the two curves.
02
Close an exit
Choose An upper exit closes and inspect the late-time channel probabilities.
What to observe: The upper asymptote requires enough total energy. A finite-width quantum packet contains an energy distribution; it can leave a small high-energy tail.
03
Separate scatter and bias
Open Ensemble and seed, increase Trajectories and save several independent seeds. Compare the final estimates with their standard errors.
What to observe: Repeated estimates fluctuate less when more trajectories are used. The gap between each method and the quantum result need not shrink.

Three Ways Through an Avoided Crossing

Follow the probability into the exits

Four exit probabilities

Population reaching the right exit