Semiclassical Theory of Inelastic Collisions. II. Momentum-Space Formulation

Delos, J. B.; Thorson, Walter R.

doi:10.1103/physreva.6.720

Cited by 161 publications

(59 citation statements)

References 15 publications

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“…• The Ehrenfest method [15][16][17][18][19] is the equivalent of the surface hopping method in the single-set formalism: each independent classical trajectory follows the gradient of a superposition of electronic adiabatic states, i.e. evolves on an effective potential energy surface.…”

Section: Introductionmentioning

confidence: 99%

Direct methods for non-adiabatic dynamics: connecting the single-set variational multi-configuration Gaussian (vMCG) and Ehrenfest perspectives

2016

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coordinates, respectively. The Hamiltonian operator reads: Ĥ (r, R) =T n (R) +T e (r) + V n−e (r, R) =T n (R) +Ĥ e (r, R) , with T n (R) the kinetic energy operator of the nuclei, T e (r) the kinetic energy operator of the electrons, V n−e (r, R) which includes all inter-particles interactions (electron-electron, nucleus-nucleus and electron-nucleus) and Ĥ e (r, R) =T e (r) + V n−e (r, R) the electronic Hamiltonian for fixed nuclei R. Unfortunately, this equation is impossible to solve analytically for more than two particles, i.e. the hydrogen atom. The field of theoretical chemistry is thus dominated by developments of numerical and approximate methods, which can be used to treat other atoms and molecules.Conventional Born-Oppenheimer molecular dynamics only allows one to simulate nuclear motion on a single potential energy surface and therefore does not describe non-adiabatic processes involving non-radiative electronic transitions. Standard grid-based quantum mechanical simulations are expensive computationally because of their exponential scaling with the system size. A separate bottleneck is the computation and fitting of the potential energy surfaces prior to any dynamics calculation. Reducing the number of nuclear degrees of freedom of the system is sometimes done to make the calculation feasible, but the validity of this approximation is limited [2][3][4]. "On-the-fly" dynamics methods have been developed to address these issues. They are referred to as direct dynamics methods since the potential energy surfaces are calculated as needed along nuclear trajectories, thus avoiding the precomputation of globally fitted surfaces, and sampling only the relevant regions of the potential energy surfaces. These nuclear trajectories are used to describe the nuclear wave packet motion, i.e. the nuclear wave packet is expanded in the basis of nuclear trajectories via expansion coefficients.Abstract In this article, we outline the current state-of-theart "on-the-fly" methods for non-adiabatic dynamics, highlighting the similarities and differences between them. We derive the equations of motion for both the Ehrenfest and variational multi-configuration Gaussian (vMCG) methods from the Dirac-Frenkel variational principle. We explore the connections between these two methods by presenting an alternative derivation of the vMCG method, which gives the Ehrenfest equations of motion when taking the appropriate limits.

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Section: Introductionmentioning

confidence: 99%

Direct methods for non-adiabatic dynamics: connecting the single-set variational multi-configuration Gaussian (vMCG) and Ehrenfest perspectives

2016

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“…The resonant conversion of waves in nonuniform media is a common phenomenon in physics and has been studied in fields as diverse as rf heating of fusion plasmas, 6 ionospheric physics, 7 ocean waves, [8][9][10] magnetohelioseismology, 11 atomic, molecular, and optical physics, [12][13][14] and neutrino physics. 15 While there is a large physics literature on conversion in one dimension ͑see Refs.…”

Section: Introductionmentioning

confidence: 99%

Local fields for asymptotic matching in multidimensional mode conversion

2007

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The problem of resonant mode conversion in multiple spatial dimensions is considered. Using phase space methods, a complete theory is developed for constructing matched asymptotic expansions that fit incoming and outgoing WKB solutions. These results provide, for the first time, a complete and practical method for including multidimensional conversion in ray tracing algorithms. The paper provides a self-contained description of the following topics: (1) how to use eikonal (also known as ray tracing or WKB) methods to solve vector wave equations and how to detect conversion regions while following rays; (2) once conversion is detected, how to fit to a generic saddle structure in ray phase space associated with the most common type of conversion; (3) given the saddle structure, how to carry out a local projection of the full vector wave equation onto a local two-component normal form that governs the two resonantly interacting waves. This determines both the uncoupled dispersion functions and the coupling constant, which in turn determine the uncoupled WKB solutions; (4) given the normal form of the local two-component wave equation, how to find the particular solution that matches the amplitude, phase, and polarization of the incoming ray, to the amplitude, phase, and polarization of the two outgoing rays: the transmitted and converted rays.

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“…The Ehrenfest method has been extensively discussed in the literature [16][17][18][19][20][21][22][23][24][25][26]. In this section, we review the Ehrenfest formalism following the elegant derivation of Tully [27].…”

Section: The Ehrenfest Approach: General Theoretical Developmentmentioning

confidence: 99%

The second-order Ehrenfest method

et al. 2014

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This article describes the Ehrenfest method and our second-order implementation (with approximate gradient and Hessian) within a CASSCF formalism. We demonstrate that the second order implementation with the predictor-corrector integration method improves the accuracy of the simulation significantly in terms of energy conservation. Although the method is general and can be used to study any coupled electron-nuclear dynamics, we apply it to investigate charge migration upon ionization of small organic molecules, focusing on benzene cation. Using this approach, we can study the evolution of a non-stationary electronic wavefunction for fixed atomic nuclei, and where the nuclei are allowed to move, to investigate the interplay between them for the first time. Analysis methods for the interpretation of the electronic and nuclear dynamics are suggested: we monitor the electronic dynamics by calculating the spin density of the system as a function of time.

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Semiclassical Theory of Inelastic Collisions. II. Momentum-Space Formulation

Cited by 161 publications

References 15 publications

Direct methods for non-adiabatic dynamics: connecting the single-set variational multi-configuration Gaussian (vMCG) and Ehrenfest perspectives

Direct methods for non-adiabatic dynamics: connecting the single-set variational multi-configuration Gaussian (vMCG) and Ehrenfest perspectives

Local fields for asymptotic matching in multidimensional mode conversion

The second-order Ehrenfest method

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