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1. Classical and nonclassical effects in surface hopping methodology for simulating coupled electronic-nuclear dynamics

   https://doi.org/10.1039/C9FD00042A  

   Martens, Craig C. (January 2019, Faraday Discussions)

   In this paper, we analyze the detailed quantum-classical behavior of two alternative approaches to simulating molecular dynamics with electronic transitions: the popular fewest switches surface hopping (FSSH) method, introduced by Tully in 1990 [Tully, \textit{J.~Chem.~Phys.}, 1990, \textbf{93}, 1061] and our recently developed quantum trajectory surface hopping (QTSH) method [Martens, \textit{J.~Phys.~Chem.~A}, 2019 \textbf{123}, 1110]. Both approaches employ an independent ensemble of trajectories that undergo stochastic transitions between electronic surfaces. The methods differ in their treatment of energy conservation, with FSSH imposing conservation of the classical kinetic plus potential energy by rescaling of the classical momenta when a surface hop occurs, while QTSH incorporates quantum forces throughout the dynamics which lead naturally to the conservation of the full quantum-classical energy. We investigate the population transfer and energy budget of the surface hopping methods for several simple model systems and compare with exact quantum result. In addition, the detailed dynamics of the trajectory ensembles in phase space are compared with the quantum evolution in the Wigner representation. Conclusions are drawn. 

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2. The quantum trajectory‐guided adaptive Gaussian methodology in the Libra software package

   https://doi.org/10.1002/qua.27078  

   Dutra, Matthew; Garashchuk, Sophya; Akimov, Alexey V. (December 2022, International Journal of Quantum Chemistry)

   Abstract In this paper, we report an implementation of the quantum trajectory‐guided adaptive Gaussian (QTAG) method in a modular open‐source Libra package for quantum dynamics calculations. The QTAG method is based on a representation of wavefunctions in terms of a quantum trajectory‐guided adaptable Gaussians basis and is generalized for time‐propagation on multiple coupled surfaces to be applicable to model nonadiabatic dynamics. The potential matrix elements are evaluated within either the local harmonic or bra‐ket‐average (linear) approximations to the potential energy surfaces, the latter being a more practical option. Performance of the QTAG method is demonstrated and discussed for the Holstein and Tully models, which are the standard benchmarks for method development in the area of nonadiabatic dynamics. 

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3. Two-way wave–vortex interactions in a Lagrangian-mean shallow water model

   https://doi.org/10.1017/jfm.2022.889  

   Maitland-Davies, Cai; Bühler, Oliver (January 2023, Journal of Fluid Mechanics)

   We derive and investigate numerically a reduced model for wave–vortex interactions involving non-dispersive waves, which we study in a two-dimensional shallow water system with an eye towards applications in atmosphere–ocean fluid dynamics. The model consists of a coupled set of nonlinear partial differential equations for the Lagrangian-mean velocity and the wave-related pseudomomentum vector field defined in generalized Lagrangian-mean theory. It allows for two-way interactions between the waves and the balanced flow that is controlled by the distribution of Lagrangian-mean potential vorticity, and for strong solutions it features a desirable exact energy conservation law for the sum of wave energy and mean flow energy. Our model goes beyond standard ray tracing as we can derive weak solutions that contain discontinuities in the pseudomomentum field, using the theory of weakly hyperbolic systems. This allows caustics to form without predicting infinite wave amplitudes, as would be the case in the standard ray-tracing theory. Suitable wave forcing and dissipation terms are added to the model and a numerical scheme for the model is implemented as a coupled set of pseudo-spectral and finite-volume integrators. Idealized examples of interactions between wavepackets and simple vortex structures are presented to illustrate the model dynamics. The unforced and non-dissipative simulations suggest a heuristic rule of ‘greedy’ waves, i.e. in the long run the wave field always extracts energy from the mean flow. 

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4. Nonadiabatic dynamics with classical trajectories: The problem of an initial coherent superposition of electronic states

   https://doi.org/10.1063/5.0186984  

   Villaseco_Arribas, Evaristo; Maitra, Neepa T; Agostini, Federica (February 2024, The Journal of Chemical Physics)

   Advances in coherent light sources and development of pump–probe techniques in recent decades have opened the way to study electronic motion in its natural time scale. When an ultrashort laser pulse interacts with a molecular target, a coherent superposition of electronic states is created and the triggered electron dynamics is coupled to the nuclear motion. A natural and computationally efficient choice to simulate this correlated dynamics is a trajectory-based method where the quantum-mechanical electronic evolution is coupled to a classical-like nuclear dynamics. These methods must approximate the initial correlated electron–nuclear state by associating an initial electronic wavefunction to each classical trajectory in the ensemble. Different possibilities exist that reproduce the initial populations of the exact molecular wavefunction when represented in a basis. We show that different choices yield different dynamics and explore the effect of this choice in Ehrenfest, surface hopping, and exact-factorization-based coupled-trajectory schemes in a one-dimensional two-electronic-state model system that can be solved numerically exactly. This work aims to clarify the problems that standard trajectory-based techniques might have when a coherent superposition of electronic states is created to initialize the dynamics, to discuss what properties and observables are affected by different choices of electronic initial conditions and to point out the importance of quantum-momentum-induced electronic transitions in coupled-trajectory schemes. 

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5. Implicit Balancing and Regularization: Generalization and Convergence Guarantees for Overparameterized Asymmetric Matrix Sensing

   Soltanolkotabi, Mahdi; Stöger, Dominik; Xie, Changzhi (July 2023, Proceedings of Thirty Sixth Conference on Learning Theory)

   Recently, there has been significant progress in understanding the convergence and generalization properties of gradient-based methods for training overparameterized learning models. However, many aspects including the role of small random initialization and how the various parameters of the model are coupled during gradient-based updates to facilitate good generalization, remain largely mysterious. A series of recent papers have begun to study this role for non-convex formulations of symmetric Positive Semi-Definite (PSD) matrix sensing problems which involve reconstructing a low-rank PSD matrix from a few linear measurements. The underlying symmetry/PSDness is crucial to existing convergence and generalization guarantees for this problem. In this paper, we study a general overparameterized low-rank matrix sensing problem where one wishes to reconstruct an asymmetric rectangular low-rank matrix from a few linear measurements. We prove that an overparameterized model trained via factorized gradient descent converges to the low-rank matrix generating the measurements. We show that in this setting, factorized gradient descent enjoys two implicit properties: (1) coupling of the trajectory of gradient descent where the factors are coupled in various ways throughout the gradient update trajectory and (2) an algorithmic regularization property where the iterates show a propensity towards low-rank models despite the overparameterized nature of the factorized model. These two implicit properties in turn allow us to show that the gradient descent trajectory from small random initialization moves towards solutions that are both globally optimal and generalize well. 

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