# Dissociation dynamics in a spatially nonuniform optical field

## Background

An optical field confined near a nanostructure can vary over a molecular
length scale. A molecule in such a field need not obey the same effective
couplings as it does under a spatially uniform dipole field. Consider a
molecular hydrogen ion aligned with a linearly polarized field. The electronic
coordinate $z$ is measured from the nuclear midpoint, and $R$ is the
internuclear separation.

## Research problem

Determine whether, and by what physically distinguishable route, spatial field
nonuniformity changes the dissociation dynamics of $\mathrm{H}_2^+$ relative
to uniform-field excitation in the model and cases below. Establish which
claims about the dissociation pathway are supported by your computed evidence,
how the conclusion depends on photon energy, and what scope that evidence
supports.

How you construct and substantiate the argument is for you to decide. Do not
assume that the nonuniform case must produce a distinct mechanism.

## Physical model and inputs

Use atomic units for the Hamiltonian and propagation. The reduced
non-Born-Oppenheimer model retains the electronic coordinate $z$ and nuclear
coordinate $R$:

$$
\begin{aligned}
\hat H_0(z,R)={}&
-\frac{1}{2\mu_z}\frac{\partial^2}{\partial z^2}
-\frac{1}{2\mu_R}\frac{\partial^2}{\partial R^2} \\
&-\frac{1}{\sqrt{(z-R/2)^2+\alpha}}
-\frac{1}{\sqrt{(z+R/2)^2+\alpha}}
+\frac{1}{\sqrt{R^2+\beta}},
\end{aligned}
$$

where $\mu_R=m_p/2$, $\mu_z=2m_pm_e/(2m_p+m_e)$,
$\alpha=1.0$, and $\beta=0.03$. Start from the field-free vibrational ground
state on the bound electronic ground-state surface.

Propagate

$$
i\frac{\partial\Psi}{\partial t}
=\left[\hat H_0+\hat V_{\mathrm{ext}}(t)\right]\Psi,
\qquad
\hat V_{\mathrm{ext}}(z,R,t)
=E_0\left[z(1+\epsilon z)-\epsilon R^2\right]
\sin(\omega t)f(t).
$$

Here $E_0$ is the field amplitude at the nuclear midpoint. This interaction
corresponds to the first-order spatial field model
$E(z)=E_0(1+2\epsilon z)$. Higher spatial orders are outside the task. The
envelope $f(t)$ rises from zero to unity with a sine-squared profile between 0
and 10 fs. It remains unity from 10 to 30 fs, then falls to zero with a
sine-squared profile at 40 fs. It is zero outside that interval. Convert photon
energy and peak intensity to $\omega$ and $E_0$ consistently, and state the
conventions and constants used.

Use both physical case sets:

| Case set | Photon energy | Peak intensity | Field nonuniformity |
|---|---|---|---|
| Nonuniformity survey | $2.0$, $3.6$, and $8.0$ eV | $1.0\times10^{12}\,\mathrm{W/cm^2}$ | Include $\epsilon=0$ and choose a justified sampling over $0<\epsilon\leq0.1\,\mathrm{a.u.}^{-1}$ that can resolve the leading response. |
| Dynamical comparison | $3.6$ eV | $3.0\times10^{13}\,\mathrm{W/cm^2}$ | Compare $\epsilon=0$ and $\epsilon=0.1\,\mathrm{a.u.}^{-1}$. |

Treat dissociation as outgoing nuclear probability at asymptotically large
separation, not as electronic ionization or undifferentiated norm loss. A
converged absorber, nuclear-flux, or physically equivalent asymptotic
construction is acceptable when its meaning is documented. Propagate long
enough to support the reported claim or quantify the unresolved transient.

## Method and execution policy

The central calculation must solve the stated two-coordinate,
electron-nuclear TDSE. A fixed-nuclei, single-surface, or rate-only surrogate
may be used as a control but cannot replace that calculation. You may choose
the propagator, eigensolver, grids, boundaries, absorbers, sampling, and
diagnostics. Declare those choices and show that the evidence used for each
material claim is numerically trustworthy. No particular solver, grid,
time-step sequence, state projection, plot layout, or validation procedure is
prescribed.

The available CPU allocation, candidate active deadline, node, network,
memory and scratch limits are specified in the accompanying
`resource_limits.json`. Complete the derivation, implementation,
calculation, validation, interpretation and final-file preparation within these
resource limits. No separate aggregate CPU-hour limit or additional task-specific RAM ceiling applies. The work must be reproducible offline after
submission.

## Submission

Submit one self-contained scientific report and an offline calculation
package. Include the code, the numerical data underlying every material claim,
 a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge, and concise environment and run information.
The report organization and supporting evidence are your choice. No fixed
number of figures, file tree, filename scheme, or complete exploratory history
is required.

## Reference and integrity policy

This examination is conducted without network access or live literature
retrieval. You may rely on your existing general scientific knowledge and, if any are
explicitly supplied with the examination, on those offline materials or
software documentation. Record every such source actually consulted. For formulas,
methods, constants, or conventions used from general knowledge rather than a
consulted source, state enough of the method basis, definition, value, units,
and convention to make the calculation auditable; do not invent a citation.
You are not required to identify or prove a publication date or to retrieve
literature during the examination.

The hidden target article and its identifying metadata, figures or captions,
supplementary material, author code or data, target-specific reproduction
code, solution or scoring material, online solutions, and worked
reproductions of this case must not be used. All submitted scientific figures
and numerical data must be generated by your own implementation of the stated
problem.

See [submission requirements](../../submission_requirements.md) for the common submission entry points.