Vibration, splitting and escape

When a Molecular Bond Breaks

Why can a kicked molecular bond break when an ideal spring of the same stiffness always pulls back? Because a real bond weakens as it stretches. Near its resting length it acts like a spring, but the pull fades with distance and the well has a finite depth. A sudden kick gives the vibration a spread of energies: the part above the well depth escapes and the bond breaks, and the rest swings back. A spring's wall keeps rising, so every part returns.

Kick a bound vibrational wavepacket into motion. Watch part of it return while another part escapes along a Morse potential, then compare the same preparation with an ideal spring.

Model assumptions

The model has one dimensionless nuclear coordinate moving on a fixed electronic potential. The initial state is the Morse ground state given a momentum kick. No laser pulse is calculated, and no particular molecule is calibrated. The probability found beyond a finite stretch is not the asymptotic dissociation yield.

How part of a wavepacket escapes a bond

PROBABILITY BEYOND STRETCH SIX

0\%

EVOLUTION TIME

\tau=0

Read the probability

What stays. What leaves.

Every instant on one map

Time runs downward. Drag the white cursor. Gold marks stretch six; it is an observation boundary, not a detector or a wall.

01.5

A probability profile

Cyan is the current profile, gray the initial state and gold the pinned prediction. The vertical scale adapts to the data.

ENERGY DRIFT

0

NORM DRIFT

0

EDGE WEIGHT

0

HIGH-FREQUENCY WEIGHT

0

Independent simulated measurements

Pick an instant, then measure freshly prepared molecules

Sample one stretch coordinate from each of 768 independently prepared molecules. The histogram comes only from those records; its axis includes every sample.

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}i\partial_\tau\psi=H\psi,\quad H=-\tfrac12\partial_x^2+V(x)\\V(x)=D(1-e^{-ax})^2\\V_{\mathrm{spring}}(x)=Da^2x^2\end{gathered}

The model starts from the exact Morse ground state and multiplies it by a momentum-kick phase. As an effective vibration model, the stretch coordinate runs over the whole real line. Rotation, electronic transitions, pulse preparation and environmental loss are left out. The far-region fraction is the probability of being far out at a given time; it is not a projection onto continuum states.

\begin{gathered}\psi(x,0)\propto z^{\lambda-1/2}e^{-z/2}e^{ip_0x}\\z=2\lambda e^{-ax},\quad\lambda=\sqrt{2D}/a\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: Why can a kicked molecular bond break when an ideal spring of the same stiffness always pulls back?
One-sentence intuition: Because the bond is anharmonic, the outer wall of the potential flattens out. A kicked bound state contains many energies, so part of the wave packet can escape while another part returns.

Core mathematical model

Hamiltonian

H=-\frac12\partial_x^2+D(1-e^{-ax})^2

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.

Bound energy levels

E_n=a\sqrt{2D}(n+\tfrac12)-\frac{a^2}{2}(n+\tfrac12)^2

There are only finitely many bound levels. These energies come from the analytic formula as a reference, not from peaks found in the position samples.

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 has one dimensionless nuclear coordinate moving on a fixed electronic potential. The initial state is the Morse ground state given a momentum kick. No laser pulse is calculated, and no particular molecule is calibrated. The probability found beyond a finite stretch is not the asymptotic dissociation yield.

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: A stronger kick sends more probability outward. An ideal spring brings it back.
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.
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.

When a Molecular Bond Breaks

What stays. What leaves.

Every instant on one map

A probability profile

Pick an instant, then measure freshly prepared molecules