cases/case14/task.md

# Two-centre structure in nondipole strong-field ionization

## Background

The finite propagation speed of a strong laser field makes longitudinal
nondipole effects accessible in photoelectron observables. A homonuclear
molecule adds two-centre coherence and orbital parity to this response. What,
if anything, can be learned about the emission process from their combination
must be established rather than assumed.

Consider a homonuclear two-centre model with laser-dressed, energetically
degenerate $\sigma_g1s$ and $\sigma_u1s$ states in a circularly polarized
infrared field. The field propagates along $+z$ and the molecular axis is

$$
\mathbf R=R(\sin\theta_m,0,\cos\theta_m).
$$

## Research problem

Determine what spatial and timing information about strong-field emission can
be inferred from the leading-order nondipole photoelectron response of this
system. Establish how the inference depends on the parity, separation and
orientation of the two-centre state, whether a common interpretation remains
valid from the compact to the widely separated regime, and under what
conditions the resulting signature is physically informative.

You must support the conclusion with your own calculation for the cases below.
How you derive the computable relations, construct the relevant observables
and organize the evidence is for you to decide.

## Physical inputs and conventions

Work in atomic units unless stated otherwise. Use the circular vector
potential

$$
\mathbf A(\eta)=-A_0(\sin\Phi,\cos\Phi,0),
\qquad \eta=t-z/c,
$$

with wavelength $3000\ \mathrm{nm}$ and total intensity
$4.0\times10^{13}\ \mathrm{W/cm^2}$. The quoted intensity is the total
intensity of the two equal circular-field quadratures. Use one consistent
relation among $\mathbf A$, the electric field and the ionization phase
$\Phi$.

Use the nondipole longitudinal light-front momentum

$$
p_-=p_z-\frac{\mathbf p^2}{2c}.
$$

The absolute ionization potential of the model dimer is not fixed. Therefore,
absolute translation of a momentum distribution is not a target. Define and
declare a consistent relative longitudinal coordinate for structural
comparisons. Any relative or dimensionless longitudinal diagnostic used in the
orientation analysis must likewise be derived and defined without assigning an
unsupported numerical value to $I_p$.

Use the source-stated tunnelling-width scale $R_w\simeq7.5$ a.u. under these
field conditions and the following physical cases:

| Case | State and geometry |
|---|---|
| Compact two-centre system | $R=5$ a.u.; laser-dressed degenerate $\sigma_g1s$ and $\sigma_u1s$ states; all inequivalent orientations from $\theta_m=0$ to $\pi/2$ |
| Widely separated system | $R=100$ a.u.; the same two parity states at $\theta_m=0$ |

For orientation-dependent observables, use the optical-phase-averaged response
of the stated circular field. Relative ionization yields and distribution
shapes may be normalized; declare the convention and do not use independent
rescalings to support a relative-magnitude claim. Absolute ionization yield is
not a grading target.

## Model boundary

Use a standard leading-order nondipole PPT/SFA treatment with the
laser-dressed degenerate two-centre $1s$ LCAO as the initial state. A Wigner or
physically equivalent phase-space representation may be used. Derive the
task-specific molecular and nondipole relations needed for your investigation
rather than importing a worked reproduction.

A full Coulomb-corrected molecular SFA calculation, finite-pulse absolute rate,
continuum-trajectory propagation and an atomic comparison system are not
required. Such extensions may be included only as clearly separated
exploratory work and receive no additional credit.

Numerical grids, integration rules, sampling, tolerances, plotting domains and
display choices are yours. You are responsible for ensuring that the computed
evidence is reliable and reproducible enough for the claims made; no particular
validation procedure is prescribed.

## Submission

Submit one self-contained scientific report and an auditable offline
calculation package. Include the task-specific derivation, code, numerical data
underlying all material claims, a provenance note listing every offline source or software document actually consulted and the method basis for items supplied from general knowledge, and concise run and
environment information. The organization of the report and the number and
form of supporting figures are yours. No fixed file tree, filename scheme,
complete exploratory history or author-style panel layout is required.

The active examination resources and limits are defined in the companion
`resource_limits.json`. All retained production calculations must run
within that fixed allocation and active wall time. More expensive modelling
does not receive additional credit.

## 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.