cases/case32/task.md

# Diagnosing coupled electron-nuclear dynamics through molecular high-harmonic spectra

## Background and physical system

High-harmonic radiation from a light molecule can carry information about
electronic motion and the accompanying nuclear dynamics. In a finite pulse,
the position and shape of a harmonic line can also depend on when within the
pulse the radiation is emitted. The question is whether this frequency-domain
information can be used as a quantitative probe of nuclear motion rather than
being interpreted only as a change in harmonic yield.

Work in atomic units with a one-electron homonuclear molecular ion aligned
with a linearly polarized laser field. Retain the internuclear coordinate
$R$ and the electronic cylindrical coordinates $(\rho,z)$; azimuthal
dependence is removed by cylindrical symmetry. After separating the
center-of-mass motion, use

$$
i\frac{\partial}{\partial t}\Psi(R,\rho,z,t)
=\left[-\frac{1}{2\alpha}\frac{\partial^2}{\partial R^2}
-\frac{\nabla_e^2}{2\beta}+V_c(R,\rho,z)-V_I(R,z,t)\right]
\Psi(R,\rho,z,t),
$$

where

$$
\nabla_e^2=\frac{1}{\rho}\frac{\partial}{\partial\rho}
\left(\rho\frac{\partial}{\partial\rho}\right)
+\frac{\partial^2}{\partial z^2},
$$

$$
V_c=\frac{Z_1Z_2}{R}
-\frac{Z_1}{\sqrt{\rho^2+(z+R/2)^2}}
-\frac{Z_2}{\sqrt{\rho^2+(z-R/2)^2}},
$$

$$
V_I=\left[
\frac{(M_1-M_2)R}{M_1+M_2}
-\frac{(2+M_1+M_2)z}{1+M_1+M_2}
\right]E(t),
$$

with

$$
\alpha=\frac{M_1M_2}{M_1+M_2},
\qquad
\beta=\frac{M_1+M_2}{M_1+M_2+1},
\qquad Z_1=Z_2=1.
$$

Treat both $\mathrm{H}_2^+$ and $\mathrm{D}_2^+$, taking each deuteron to
have twice the proton mass, and initialize each moving-nuclei calculation in
the corresponding field-free ground state. Also construct a fixed-nuclei
$\mathrm{H}_2^+$ reference at the field-free equilibrium geometry of the
hydrogen molecular ion.

Use an 800 nm pulse of peak intensity
$3.0\times10^{14}\ \mathrm{W\,cm^{-2}}$. With the field maximum at $t=0$,
take

$$
E(t)=E_0\cos^2\left(\frac{\pi t}{T}\right)\cos(\omega t),
\qquad |t|\leq \frac{T}{2},
$$

and $E(t)=0$ otherwise, where $T$ is 20 optical cycles and
$\omega=2\pi c/(800\ \mathrm{nm})$.

## Research problem

Determine whether frequency modulation of the molecular high-harmonic
response provides a defensible probe of nuclear motion in this system. Use
the fixed-nuclei calculation and isotope substitution to test the physical
interpretation, and determine what aspects of the nuclear dynamics can and
cannot be inferred from the calculated radiation.

The production calculation must retain the full $(R,\rho,z)$
non-Born-Oppenheimer Hamiltonian stated above. A reduced-dimensional or soft-core
model may be discussed only as a clearly separated exploratory calculation;
it cannot replace the required evidence.

The numerical representation, propagation algorithm, grids or bases, time
step, boundaries, outgoing-wave treatment, transform and time-frequency
procedures, estimators, normalization, and evidence architecture are for you
to choose. State the choices that materially affect your claims and provide
evidence sufficiently reliable and reconstructible to support them. Absolute
spectral intensity, a particular display range, and reproduction of a fixed
panel layout are not grading targets.

Do not add electronic correlation beyond the stated one-electron model,
rotational dynamics, macroscopic propagation, focal or orientational
averaging, phenomenological ionization rates, or modified molecular
potentials to the required production calculation. Clearly separated
extensions are allowed but cannot replace the stated model.

## Submission and resources

Submit one self-contained scientific report and an auditable offline
calculation package containing 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. No fixed file tree, panel count, filename
scheme, or complete exploration history is required.

Resource limits are specified exclusively by the accompanying
`resource_limits.json`: its fixed node count, CPU cores per node,
work time and closed-network boundary apply. No separate aggregate CPU-hour limit applies. Greater computational
expense receives no 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.