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Recent experiments on the return to equilibrium of solutions of entangled polymers stretched by extensional flows [Zhou and Schroeder, Phys. Rev. Lett. 120, 267801 (2018)] have highlighted the possible role of the tube model's two-step mechanism in the process of chain relaxation. read more In this paper, motivated by these findings, we use a generalized Langevin equation (GLE) to study the time evolution, under linear mixed flow, of the linear dimensions of a single finitely extensible Rouse polymer in a solution of other polymers. Approximating the memory function of the GLE, which contains the details of the interactions of the Rouse polymer with its surroundings, by a power law defined by two parameters, we show that the decay of the chain's fractional extension in the steady state can be expressed in terms of a linear combination of Mittag-Leffler and generalized Mittag-Leffler functions. For the special cases of elongational flow and steady shear flow, and after adjustment of the parameters in the memory function, our calculated decay curves provide satisfactory fits to the experimental decay curves from the work of Zhou and Schroeder and earlier work of Teixeira et al. [Macromolecules 40, 2461 (2007)]. The non-exponential character of the Mittag-Leffler functions and the consequent absence of characteristic decay constants suggest that melt relaxation may proceed by a sequence of steps with an essentially continuous, rather than discrete, spectrum of timescales.Recent work shows that strong stability and dimensionality freedom are essential for robust numerical integration of thermostatted ring-polymer molecular dynamics (T-RPMD) and path-integral molecular dynamics, without which standard integrators exhibit non-ergodicity and other pathologies [R. Korol et al., J. Chem. Phys. 151, 124103 (2019) and R. Korol et al., J. Chem. Phys. 152, 104102 (2020)]. In particular, the BCOCB scheme, obtained via Cayley modification of the standard BAOAB scheme, features a simple reparametrization of the free ring-polymer sub-step that confers strong stability and dimensionality freedom and has been shown to yield excellent numerical accuracy in condensed-phase systems with large time steps. Here, we introduce a broader class of T-RPMD numerical integrators that exhibit strong stability and dimensionality freedom, irrespective of the Ornstein-Uhlenbeck friction schedule. In addition to considering equilibrium accuracy and time step stability as in previous work, we evaluate the integrators on the basis of their rates of convergence to equilibrium and their efficiency at evaluating equilibrium expectation values. Within the generalized class, we find BCOCB to be superior with respect to accuracy and efficiency for various configuration-dependent observables, although other integrators within the generalized class perform better for velocity-dependent quantities. Extensive numerical evidence indicates that the stated performance guarantees hold for the strongly anharmonic case of liquid water. Both analytical and numerical results indicate that BCOCB excels over other known integrators in terms of accuracy, efficiency, and stability with respect to time step for practical applications.Tip-enhanced Raman spectroscopy in combination with scanning tunneling microscopy could produce ultrahigh-resolution Raman spectra and images for single-molecule vibrations. Furthermore, a recent experimental study successfully decoupled the interaction between the molecule and the substrate/tip to investigate the intrinsic properties of molecules and their near-field interactions by Raman spectroscopy. In such a circumstance, more explicit treatments of the near field and molecular interactions beyond the dipole approximation would be desirable. Here, we propose a theoretical method based on the multipolar Hamiltonian that considers full spatial distribution of the electric field under the framework of real-time time-dependent density functional theory. This approach allows us to treat the on- and off-resonance Raman phenomena on the same footing. For demonstration, a model for the on- and off-resonance tip-enhanced Raman process in benzene was constructed. The obtained Raman spectra are well understood by considering both the spatial structure of the near field and the molecular vibration in the off-resonance condition. For the on-resonance condition, the Raman spectra are governed by the transition moment, in addition to the selection rule of off-resonance Raman. Interestingly, on-resonance Raman can be activated even when the near field forbids the π-π* transition at equilibrium geometry due to vibronic couplings originating from structural distortions.Microkinetic modeling has drawn increasing attention for quantitatively analyzing catalytic networks in recent decades, in which the speed and stability of the solver play a crucial role. However, for the multi-step complex systems with a wide variation of rate constants, the often encountered stiff problem leads to the low success rate and high computational cost in the numerical solution. Here, we report a new efficient sensitivity-supervised interlock algorithm (SSIA), which enables us to solve the steady state of heterogeneous catalytic systems in the microkinetic modeling with a 100% success rate. In SSIA, we introduce the coverage sensitivity of surface intermediates to monitor the low-precision time-integration of ordinary differential equations, through which a quasi-steady-state is located. Further optimized by the high-precision damped Newton's method, this quasi-steady-state can converge with a low computational cost. Besides, to simulate the large differences (usually by orders of magnitude) among the practical coverages of different intermediates, we propose the initial coverages in SSIA to be generated in exponential space, which allows a larger and more realistic search scope. On examining three representative catalytic models, we demonstrate that SSIA is superior in both speed and robustness compared with its traditional counterparts. This efficient algorithm can be promisingly applied in existing microkinetic solvers to achieve large-scale modeling of stiff catalytic networks.