Exponential Time Differencing-Padé Finite Element Method for Nonlinear Convection-Diffusion-Reaction Equations with Time Constant Delay

In this paper, ETD3-Padé and ETD4-Padé Galerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions. An ETD-based RK is used for time integration of the corresponding equation. To overcome a well-known difficulty of numerical instability associated with the computation of the exponential operator, the Padé approach is used for such an exponential operator approximation, which in turn leads to the corresponding ETD-Padé schemes. An unconditional $L^2$ numerical stability is proved for the proposed numerical schemes, under a global Lipshitz continuity assumption. In addition, optimal rate error estimates are provided, which gives the convergence order of $O(k^{3}+h^{r})$ (ETD3-Padé) or $O(k^{4}+h^{r})$ (ETD4-Padé) in the $L^2$ norm, respectively. Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.     

Computational studies on the excited state decay rates in aggregates of two-coordinate Cu (I) complexes: Thermally Activated Delayed Fluorescence and Aggregation Induced Emis

Two-coordinate Cu (I) complexes have attracted great interest recently because of the rich photophysical property in solid state, including the aggregation-induced thermal activated delayed fluorescence. Here, we summarize our theoretical investigations on the excited state structure and decay dynamics for the two-coordinate Cu (I) complexes in solution phase and solid state by the thermal vibration correlation function rate formalism we developed earlier coupled with time-dependent density-functional theory within polarizable continuum model and hybrid quantum and molecular mechanics. First, for the CAAC Cu (I) Cl complex, we found that the nature of the excited state undergoes a change from metal-to-ligand charge transfer (MLCT) in solution to hybrid halogen-to-ligand charge transfer and MLCT in solid state. The bending vibrations of the C Cu Cl and Cu C N bonds are restricted in aggregates, reducing the non-radiative decay rate to cause strong solid-state fluorescence. Second, for CAAC Cu (I) Cz, we found that both intersystem crossing (ISC) and reverse intersystem crossing (rISC) are enhanced by 2–4 orders of magnitudes upon aggregation, leading to highly efficient thermally activated delayed fluorescence (TADF). The enhanced ISC and rISC rates can be attributed to the increase of the metal proportion in the frontier molecular orbitals, leading to an enhanced spin−orbit coupling between S₁ and T_1. The reaction barriers for ISC and rISC are much lower in solution than that in aggregate phase resulting in a decrease in energy gap $∆E_{\mathrm{ST}}$ and an increase in the relative reorganization energy through bending the angle ∠C − Cu − N for T₁. Our theoretical studies provide a clear rationalization for the highly efficient solid-state luminescence character of two-coordinate Cu (I) complexes and may clarify the ongoing dispute on the understanding of the high TADF quantum efficiency.