High-precision evaluation of Wigner's d matrix by exact diagonalization

Controllable Frequency Dependence of Resonance Energy Transfer Coupled with Localized Surface Plasmon Polaritons

We investigate the intrinsic characteristics of resonance energy transfer (RET) coupled with localized surface plasmon polaritons (LSPPs) from the perspective of macroscopic quantum electrodynamics. To quantify the effect of LSPPs, we propose a numerical scheme that allows us to accurately calculate the rate of RET between a donor-acceptor pair near a nanoparticle. Our study shows that LSPPs can be used to enhance the RET rate significantly and control its frequency dependence by modifying a core/shell structure, which indicates the possibility of RET rate optimization. Moreover, we systematically explore the angle (distance) dependence of the RET rate and analyze its origin. According to different frequency regimes, the angle dependence of RET is dominated by different mechanisms, such as LSPPs, surface plasmon polaritons (SPPs), and anti-resonance. For the proposed core/shell structure, the characteristic distance of RET coupled with LSPPs (approximately 0.05 emission wavelength) is shorter than that of RET coupled with SPPs (approximately 0.1 emission wavelength), which may provide promising applications in energy science.

Wigner numbers

All reduced Wigner rotation matrix elements d_M^'M ^J(θ) can be evaluated very efficiently as a linear combination of either cos(Nθ) or sin(Nθ) terms as N runs in unit steps from either 0 or 12 to J. Exact, infinite-precision formulas are derived here for the Fourier coefficients in these d_M^'M ^J(θ) expressions by finding remarkable analytic solutions for the normalized eigenvectors of arbitrarily large matrices that represent the Ĵ_Y angular momentum operator in the basis of Ĵ_Z eigenstates. The solutions involve collections of numbers W_m,n ^J for (m, n) = (J-M, J-N) ∈ [0, 2J] that satisfy the recursion relation (m+1)W_m+1,n ^J-2(J-n)W_m,n ^J+(2J-m+1)W_m-1,n ^J=0. These quantities, designated here as Wigner numbers, are proved to be integers that exhibit myriad intriguing mathematical properties, including various closed combinatorial formulas, (M, N) sum rules, three separate M-, N-, and J-recursion relations, and a large-J limiting differential equation whose applicable solutions are products of a polynomial and a Gaussian function in the variable z = -2^1/2(J + 1)^-1/2M. Accordingly, the Wigner numbers constitute a new thread of mathematics extending outside the context of their immediate discovery. In the midst of the W_m,n ^J proofs, a class of previously unknown combinatorial summation identities is also found from Wigner number orthonormalization conditions.

On the Angular Distribution of the H+Li2 Cross Sections: a Converged Time-Independent Quantum Scattering Study

A thorough time-independent quantum scattering study is performed on a benchmark potential energy surface for the H+Li₂ reaction at the fundamental electronic state. Integral and differential cross sections are calculated along with thermal rate coefficients until convergence is reached. Our findings show that vibrational and rotational excitations of the reactant hinder reactivity, though for the latter a considerable reaction promotion was spotted as we increase the reactant rotational quantum number until the critical value of j = 4. Such a promotion then begins to retract, eventually becoming an actual inhibition for larger j. In a straightforward manner, the concept of time-independent methods implemented in this study allowed this accurate state-to-state analysis. Furthermore, a nearly isotropic behaviour of the scattering is noted to take place from the angular point of view. Remarkably, our computational protocol is ideally suited to yield converged thermal rate coefficients, revealing a non-Arrhenius pattern and showing that J-shifting approach fails to describe this particular reaction. Our results, when compared to previous and independent ones, reinforce the latest theoretical reference for future validation in the experimental field.