Scientific sources - Case18
References
Lina Bielke, Samuel Schöpa, Falk-Erik Wiechmann, Franziska Fennel, and Dieter Bauer. 2026. High-harmonic generation from two weakly coupled molecules: A simple tight-binding model. Physical Review A, 113, 043106. Source.
Four-site exact and adia-intra HHG identities, weak-coupling response reorganization, transition-assisted departures and qualitative/quantitative scope. Source numerical implementation and display choices are examples, not required deliverables.
Criterion-to-source crosswalk
Locations refer to the article or named preprint/manuscript page, not a figure-file index. Source examples establish scientific identity and robust structure; the task defines required reporting, case coverage and risk-triggered validation. No original source images are reproduced.
| Criteria | Source locations, scientific use and limits |
|---|---|
| F1 | article p. 043106-2, Eqs. (1)-(8), Fig. 1(a,b) Four-site geometry, half-filled spin-degenerate occupation and the reference-gap drive define the target; frequency stays fixed during the coupling sweep. |
| F2 | article p. 043106-3, Eqs. (9)-(13) Both doubly occupied orbitals contribute coherently to the total acceleration before spectral power is formed. |
| F3 | article pp. 043106-5-6, Eqs. (14)-(20) Instantaneous eigenstates define the adia-intra position; it is not the exact time propagation. |
| O1, I1, M1 | article pp. 043106-3-5, Figs. 2, 3 and 5; paragraph following Fig. 5 The molecular-direction family is nearly coupling independent; cancellation of linear acceleration terms leaves an intermolecular yield proportional to the fourth power of weak coupling. |
| O2, I2, M3, M4 | article pp. 043106-4-5, Fig. 4(b) and Fig. 5 Order-dependent competition switches the maximizing polarization, with higher orders switching at weaker coupling. |
| O3, I3, M5 | article pp. 043106-7-8, Figs. 7-9 and adjacent comparison with Figs. 2, 4(b) and 5 Adiabatic following retains broad angular organization and ordered competition, not quantitative equality. |
| O4, I4, M2, M6, M7 | article pp. 043106-6-8, Fig. 6, Fig. 8 white comparison marks and Fig. 9; Eqs. (18)-(21) Omitted instantaneous-state transitions reduce the intermolecular response and displace crossovers toward stronger coupling. The half-frequency Fig. 6 is explanatory context, not a compulsory input case. |
| V1 | article p. 043106-5, paragraph beginning with the strong-coupling regime; p. 043106-8, Conclusion The four-level model loses quantitative interpretability at strong coupling; the task imposes the weak-coupling scope. |
| V2 | article p. 043106-3, Eqs. (10)-(13), and p. 043106-6, Eqs. (18)-(20) These definitions explain why relative scales, harmonic assignment and instantaneous-state tracking can threaten the comparison. Validation is task-defined and risk-triggered, not an extra figure obligation. |
Formula and convention notes
Source-numbered identities below are paraphrased mathematically. Explicit algebraic consequences and accepted model-conditioned representations are identified as such; they do not add a required solver, plot or validation recipe.
Total position is $\mathbf r(t)=2\langle u_0(t)|\hat{\mathbf r}|u_0(t)\rangle+2\langle u_1(t)|\hat{\mathbf r}|u_1(t)\rangle$. With $\mathbf a=\ddot{\mathbf r}$, $I(\omega)\propto|\int w(t)\mathbf a(t)e^{i\omega t}dt|^2$ and $I_n=\int_{B_n}I(\omega)d\omega$. The spin factors and coherent sum are physical; a declared equivalent window and order band $B_n$ are allowed (Eqs. (10)-(13)).
For instantaneous eigenstates $\mu_n(t)$, $\mathbf r_{mn}=\langle\mu_m|\hat{\mathbf r}|\mu_n\rangle$ and $\mathbf r_{\rm adia-intra}=2\mathbf r_{00}+2\mathbf r_{11}$ (Eq. (20)). Use the same acceleration/spectrum convention for comparison. Weak-coupling cancellation gives $\mathbf a_{\rm inter}=O(g^2)$ and $I_{\rm inter}=O(g^4)$, $g=|t_1/t_0|$; this is not a universal strong-coupling law.
Scope of the source anchors
Figure panels, displayed source coordinates, curve multipliers, numerical grids and named controls are reference examples, not compulsory submission layouts or hidden pointwise targets. Equivalent evidence must preserve the actual physical observable, case, relative scale and discriminating function. Correct data, candidate identification, causal explanation and risk resolution are separate evidential acts. Existing branch-local dependencies, applicability/N/A and descriptive Research Closure rules remain in the case rubric.