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A partially linearized spin-mapping approach for nonadiabatic dynamics. II. Analysis and comparison with related approaches

Jonathan R. Mannouch, Jeremy O. Richardson

Published Jan 1, 2020• 34 citations
Publisher

Abstract

In a previous paper [J. R. Mannouch and J. O. Richardson, J. Chem. Phys. 153, 194109 (2020)], we derived a new partially linearized mapping-based classical-trajectory technique called the spin partially linearized density matrix (spin-PLDM) approach. This method describes the dynamics associated with the forward and backward electronic path integrals using a Stratonovich–Weyl approach within the spin-mapping space. While this is the first example of a partially linearized spin-mapping method, fully linearized spin-mapping is already known to be capable of reproducing dynamical observables for a range of nonadiabatic model systems reasonably accurately. Here, we present a thorough comparison of the terms in the underlying expressions for the real-time quantum correlation functions for spin-PLDM and fully linearized spin mapping in order to ascertain the relative accuracy of the two methods. In particular, we show that spin-PLDM contains an additional term within the definition of its real-time correlation function, which diminishes many of the known errors that are ubiquitous for fully linearized approaches. One advantage of partially linearized methods over their fully linearized counterparts is that the results can be systematically improved by re-sampling the mapping variables at intermediate times. We derive such a scheme for spin-PLDM and show that for systems for which the approximation of classical nuclei is valid, numerically exact results can be obtained using only a few “jumps.” Additionally, we implement focused initial conditions for the spin-PLDM method, which reduces the number of classical trajectories that are needed in order to reach convergence of dynamical quantities, with seemingly little difference to the accuracy of the result.

DOI
10.3929/ethz-b-000452283

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In a previous paper [J. R. Mannouch and J. O. Richardson, J. Chem. Phys. 153, 194109 (2020)], we derived a new partially linearized mapping-based classical-trajectory technique called the spin partially linearized density matrix (spin-PLDM) approach. This method describes the dynamics associated with the forward and backward electronic path integrals using a Stratonovich–Weyl approach within the spin-mapping space. While this is the first example of a partially linearized spin-mapping method, fully linearized spin-mapping is already known to be capable of reproducing dynamical observables for a range of nonadiabatic model systems reasonably accurately. Here, we present a thorough comparison of the terms in the underlying expressions for the real-time quantum correlation functions for spin-PLDM and fully linearized spin mapping in order to ascertain the relative accuracy of the two methods. In particular, we show that spin-PLDM contains an additional term within the definition of its real-time correlation function, which diminishes many of the known errors that are ubiquitous for fully linearized approaches. One advantage of partially linearized methods over their fully linearized counterparts is that the results can be systematically improved by re-sampling the mapping variables at intermediate times. We derive such a scheme for spin-PLDM and show that for systems for which the approximation of classical nuclei is valid, numerically exact results can be obtained using only a few “jumps.” Additionally, we implement focused initial conditions for the spin-PLDM method, which reduces the number of classical trajectories that are needed in order to reach convergence of dynamical quantities, with seemingly little difference to the accuracy of the result.
We present a new approach for calculating quantum time correlation functions for systems whose dynamics exhibits relevant nonadiabatic effects. The method involves partial linearization of the full quantum path-integral expression for the time correlation function written in the nonadiabatic mapping Hamiltonian formalism. Our analysis gives an algorithm which is both numerically efficient and accurate as we demonstrate in test calculations on the spin-boson model where we find results in good agreement with exact calculations. The accuracy of our new approach is comparable to that of calculations performed using other approximate methods over a relatively broad range of model parameters. However, our method converges relatively quickly when compared with most alternative schemes. These findings are very encouraging in view of the application of the new method for studying realistic nonadiabatic model problems in the condensed phase.
We derive a partially linearized version of the mapping approach to surface hopping (MASH) that provides a new method for simulating nonadiabatic dynamics in molecular systems. As in the original MASH formalism, trajectories travel on adiabatic surfaces with hops determined by the dynamics of a spin vector on the Bloch sphere. However, because the new approach is based on the partially linearized density matrix (PLDM), it has two spin vectors, which may be in the same or different hemispheres of the Bloch sphere. The former scenario describes an electronic population moving on a single adiabatic surface, whereas the latter describes a coherence moving on the average surface. We show that this new method, called MASH-PLDM, leads to improved results over the original MASH dynamics for a variety of scattering problems and spin-boson models, especially in cases where coherences play an important role. A further benefit of the new methodology is that it can treat multi-time correlation functions such as those required for nonlinear spectroscopy. We demonstrate this capability with a proof-of-principle simulation of time-resolved pump-probe spectroscopy.
An approach to nonadiabatic dynamics of atoms in molecular and condensed matter systems under general nonequilibrium conditions is proposed. In this method interaction between nuclei and electrons is considered explicitly up to the second order in atomic displacements defined with respect to the mean atomic trajectory, enabling one to consider movement of atoms beyond their simple vibrations. Both electrons and nuclei are treated fully quantum-mechanically using a combination of path integrals applied to nuclei and nonequilibrium Green's functions to electrons. Our method is partitionless: initially, the entire system is coupled and assumed to be at thermal equilibrium. Then, the exact application of the Hubbard-Stratonovich transformation in mixed real and imaginary times enables us to obtain, without doing any additional approximations, an exact expression for the reduced density matrix for nuclei and hence an effective quantum Liouville equation for them, both containing Gaussian noises. It is shown that the time evolution of the expectation values for atomic positions is described by an infinite hierarchy of stochastic differential equations for atomic positions and momenta and their various fluctuations. The actual dynamics is obtained by sampling all stochastic trajectories. It is expected that applications of the method may include photoinduced chemical reactions (e.g., dissociation), electromigration, and atomic manipulation in scanning tunneling microscopy, to name just a few.
Mapping methods are often used for the numerical simulation of nonadiabatic systems by propagating classical mapping variable trajectories. A recently popularised mapping method is spin-mapping, whose mapping variables arise from quantum mechanical operators with symmetries described by a Lie-Poisson algebra. Simulating the classical-like dynamics of spin-mapping systems accurately is generally challenging, with many methods unable to preserve the underlying geometric structure of the symplectic form. The Spin-MInt algorithm is a recently proposed algorithm propagating spin-mapping variables, with a direct proof of symplecticity existing only for 2 electronic states. Here, we directly prove the symplecticity of the Spin-MInt algorithm for a general KKK electronic states. A review of the symplectic nature of coadjoint orbits of the su(K)\mathfrak{su}(K)su(K) Lie-Poisson algebra provides the framework needed to understand symplecticity of the Spin-MInt algorithm in this general case. The symplecticity of the method on the associated coadjoint orbit is then shown for what we believe to be the first time via an explicit verification of the symplecticity condition MJMT=J\mathbf{MJ}\mathbf{M}^\textrm{T}=\mathbf{J}MJMT=J exploiting the Lie-Poisson structure of the system. To our knowledge, this is the first time the monodromy matrix for the Spin-MInt algorithm has been explicitly stated using canonical coordinates on the coherent state manifold for a general number of states. We hope that this will assist the development of classical-like spin-mapping methods which might utilise elements of the monodromy matrix, and inform future work on similar symplectic algorithms for coupled and uncoupled Lie-Poisson systems.
Accurate quantum dynamics simulations of nonadiabatic processes are important for studies of electron transfer, energy transfer, and photochemical reactions in complex systems. In this comparative study, we benchmark various approximate nonadiabatic dynamics methods with mapping variables against numerically exact calculations based on the tensor-train (TT) representation of high-dimensional arrays, including TT-KSL for zero-temperature dynamics and TT-thermofield dynamics for finite-temperature dynamics. The approximate nonadiabatic dynamics methods investigated include mixed quantum-classical Ehrenfest mean-field and fewest-switches surface hopping, linearized semiclassical mapping dynamics, symmetrized quasiclassical dynamics, the spin-mapping method, and extended classical mapping models. Different model systems were evaluated, including the spin-boson model for nonadiabatic dynamics in the condensed phase, the linear vibronic coupling model for electronic transition through conical intersections, the photoisomerization model of retinal, and Tully's one-dimensional scattering models. Our calculations show that the optimal choice of approximate dynamical method is system-specific, and the accuracy is sensitively dependent on the zero-point-energy parameter and the initial sampling strategy for the mapping variables.
We study a number of filtering schemes for the reduction of the statistical error in nonadiabatic calculations by means of the quantum-classical Liouville equation. In particular, we focus on a scheme based on setting a threshold value on the sampling weights, so that when the threshold is overcome the value of the weight is reset, and on another approach which prunes the ensemble of the allowed nonadiabatic transitions according to a generalized sampling probability. Both methods have advantages and drawbacks, however, their combination drastically improves the performance of an algorithm known as the sequential short-time step propagation [MacKernan et al., J. Phys: Condens. Matter 14, 9069 (2002)], which is derived from a simple first order expansion of the quantum-classical propagator. Such an algorithm together with the combined filtering procedures produce results that compare very well with those obtained by means of numerically accurate path integral quantum calculations for the spin-boson model, even for intermediate and strong coupling regimes.