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

$$i\frac{\mathrm{\partial }}{\mathrm{\partial }t}\mathrm{\Psi }(R,\rho ,z,t)=[-\frac{1}{2\alpha }\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{R}^2}-\frac{{\mathrm{\nabla }}_e^2}{2\beta }+V_c(R,\rho ,z)-V_I(R,z,t)]\mathrm{\Psi }(R,\rho ,z,t),i\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\backslash Psi(R,\backslash rho,z,t)\; =\backslash left[-\backslash frac\{1\}\{2\backslash alpha\}\backslash frac\{\backslash partial^2\}\{\backslash partial\; R^2\}\; -\backslash frac\{\backslash nabla\_e^2\}\{2\backslash beta\}+V\_c(R,\backslash rho,z)-V\_I(R,z,t)\backslash right]\; \backslash Psi(R,\backslash rho,z,t),$$

where

$${\mathrm{\nabla }}_e^2=\frac{1}{\rho }\frac{\mathrm{\partial }}{\mathrm{\partial }\rho }\left(\rho \frac{\mathrm{\partial }}{\mathrm{\partial }\rho }\right)+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2},\backslash nabla\_e^2=\backslash frac\{1\}\{\backslash rho\}\backslash frac\{\backslash partial\}\{\backslash partial\backslash rho\}\; \backslash left(\backslash rho\backslash frac\{\backslash partial\}\{\backslash partial\backslash rho\}\backslash right)\; +\backslash frac\{\backslash partial^2\}\{\backslash partial\; z^2\},$$

$$V_c=\frac{Z_1{Z}_2}{R}-\frac{Z_1}{\sqrt{{\rho }^2+(z+R\mathrm{/}2{)}^2}}-\frac{Z_2}{\sqrt{{\rho }^2+(z-R\mathrm{/}2{)}^2}},V\_c=\backslash frac\{Z\_1Z\_2\}\{R\}\; -\backslash frac\{Z\_1\}\{\backslash sqrt\{\backslash rho^2+(z+R/2)^2\}\}\; -\backslash frac\{Z\_2\}\{\backslash sqrt\{\backslash rho^2+(z-R/2)^2\}\},$$

$$V_I=[\frac{(M_1-M_2)R}{M_1+M_2}-\frac{(2+M_1+M_2)z}{1+M_1+M_2}]E(t),V\_I=\backslash left[\; \backslash frac\{(M\_1-M\_2)R\}\{M\_1+M\_2\}\; -\backslash frac\{(2+M\_1+M\_2)z\}\{1+M\_1+M\_2\}\; \backslash right]E(t),$$

with

$$\alpha =\frac{M_1{M}_2}{M_1+M_2},\beta =\frac{M_1+M_2}{M_1+M_2+1},Z_1=Z_2=1.\backslash alpha=\backslash frac\{M\_1M\_2\}\{M\_1+M\_2\},\; \backslash qquad\; \backslash beta=\backslash frac\{M\_1+M\_2\}\{M\_1+M\_2+1\},\; \backslash qquad\; Z\_1=Z\_2=1.$$

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

$$E(t)=E_0{\cos }^2\left(\frac{\pi t}{T}\right)\cos (\omega t),\mathrm{\mid }t\mathrm{\mid }\le \frac{T}{2},E(t)=E\_0\backslash cos^2\backslash left(\backslash frac\{\backslash pi\; t\}\{T\}\backslash right)\backslash cos(\backslash omega\; t),\; \backslash qquad\; |t|\backslash leq\; \backslash frac\{T\}\{2\},$$

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 for the common submission entry points.