Asymptotic Couplings Research Articles

Treatment of asymptotic non-adiabatic couplings with higher order reprojection method in the diabatic representation.

The problem of asymptotic non-adiabatic couplings in heavy particle collisions is treated using the reprojection method. The mixing matrix that mixes the asymptotic solutions of the coupled states to obtain appropriate boundary conditions is here derived to second order, yielding a faster convergence of the cross section. In addition, the reprojection method is implemented in a diabatic representation and applied to inelastic scattering of Li + Na and H + H collisions and to mutual neutralization in H+ + H- collisions.

Asymptotic couplings by reflection and applications for nonlinear monotone SPDES

Asymptotic couplings by reflection are constructed for a class of nonlinear monotone SPDEs (stochastic partial differential equations). As applications, gradient/Hölder estimates as well as exponential convergence are derived for the associated Markov semigroup. The main results are illustrated by stochastic generalized porous media equations, stochastic p-Laplacian equations, and stochastic generalized fast-diffusion equations. We emphasize that the gradient estimate is studied for the first time for these equations.

Conditional ergodicity in infinite dimension

The goal of this paper is to develop a general method to establish conditional ergodicity of infinite-dimensional Markov chains. Given a Markov chain in a product space, we aim to understand the ergodic properties of its conditional distributions given one of the components. Such questions play a fundamental role in the ergodic theory of nonlinear filters. In the setting of Harris chains, conditional ergodicity has been established under general nondegeneracy assumptions. Unfortunately, Markov chains in infinite-dimensional state spaces are rarely amenable to the classical theory of Harris chains due to the singularity of their transition probabilities, while topological and functional methods that have been developed in the ergodic theory of infinite-dimensional Markov chains are not well suited to the investigation of conditional distributions. We must therefore develop new measure-theoretic tools in the ergodic theory of Markov chains that enable the investigation of conditional ergodicity for infinite dimensional or weak-* ergodic processes. To this end, we first develop local counterparts of zero-two laws that arise in the theory of Harris chains. These results give rise to ergodic theorems for Markov chains that admit asymptotic couplings or that are locally mixing in the sense of H. F\"{o}llmer, and to a non-Markovian ergodic theorem for stationary absolutely regular sequences. We proceed to show that local ergodicity is inherited by conditioning on a nondegenerate observation process. This is used to prove stability and unique ergodicity of the nonlinear filter. Finally, we show that our abstract results can be applied to infinite-dimensional Markov processes that arise in several settings, including dissipative stochastic partial differential equations, stochastic spin systems and stochastic differential delay equations.

Inelastic Collision Processes for Formation of Spectral Line Shapes in Stellar Atmospheres. Reprojection method

Modeling of formation of spectral lines in stellar atmospheres requires accurate data on inelastic collision processes. The only source of this information is theoretical calculations mainly performed within the standard adiabatic Born-Oppenheimer approach. It is shown that the conventional application of this approach encounters severe difficulties, in particular, due to nonzero couplings in the asymptotic region. This results in the fact that the conventional applications do not provide convergences, which are necessary for calculations of inelastic cross sections, giving infinite cross sections. The proposed remedy is the quantum reprojection method within the standard adiabatic Born-Oppenheimer approach. The method takes nonvanishing asymptotic nonadiabatic couplings into account and distinguishes asymptotic currents in molecular-state channels and in atomic-state channels. The method is demonstrated for the example of low-energy inelastic Li + Na collisions for which the conventional application of the Born-Oppenheimer approach fails.

Revised Born-Oppenheimer approach and a reprojection method for inelastic collisions

The quantum reprojection method within the standard adiabatic Born-Oppenheimer approach is derived for multielectron collision systems. The method takes nonvanishing asymptotic nonadiabatic couplings into account and distinguishes asymptotic currents in molecular state and in atomic state channels, leading to physically consistent and reliable results. The method is demonstrated for the example of low-energy inelastic Li+Na collisions, for which the conventional application of the standard adiabatic Born-Oppenheimer approach fails and leads to paradoxes such as infinite inelastic cross sections.

Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations

There are many Markov chains on infinite dimensional spaces whose one-step transition kernels are mutually singular when starting from different initial conditions. We give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of Harris’ theorem. The first uses the existence of couplings which draw the solutions together as time goes to infinity. Such “asymptotic couplings” were central to (Mattingly and Sinai in Comm Math Phys 219(3):523–565, 2001; Mattingly in Comm Math Phys 230(3):461–462, 2002; Hairer in Prob Theory Relat Field 124:345–380, 2002; Bakhtin and Mattingly in Commun Contemp Math 7:553–582, 2005) on which this work builds. As in Bakhtin and Mattingly (2005) the emphasis here is on stochastic differential delay equations. Harris’ celebrated theorem states that if a Markov chain admits a Lyapunov function whose level sets are “small” (in the sense that transition probabilities are uniformly bounded from below), then it admits a unique invariant measure and transition probabilities converge towards it at exponential speed. This convergence takes place in a total variation norm, weighted by the Lyapunov function. A second aim of this article is to replace the notion of a “small set” by the much weaker notion of a “d-small set,” which takes the topology of the underlying space into account via a distance-like function d. With this notion at hand, we prove an analogue to Harris’ theorem, where the convergence takes place in a Wasserstein-like distance weighted again by the Lyapunov function. This abstract result is then applied to the framework of stochastic delay equations. In this framework, the usual theory of Harris chains does not apply, since there are natural examples for which there exist no small sets (except for sets consisting of only one point). This gives a solution to the long-standing open problem of finding natural conditions under which a stochastic delay equation admits at most one invariant measure and transition probabilities converge to it.

Electron translation and asymptotic couplings in low-energy atomic collisions

The calculation of cross sections for inelastic and charge-exchange atomic collisions encounters a principal difficulty. The coordinates used to describe the collision event are usually not appropriate for the description of the collision partners long before and after the collision. As a consequence, the coupling matrix elements that govern inelastic transitions can remain nonzero for large internuclear distance. We present a simple procedure to deal with this situation in the framework of a full quantum-mechanical treatment. It applies to radial as well as rotational coupling mechanisms. We demonstrate the practical applicability by numerical examples.

Approach to electron translation in low-energy atomic collisions

We consider two conceptional problems that arise in the quantum-mechanical calculation of inelastic atomic collisions. The definition of the coupling matrix elements, which govern the inelastic transitions, seems to be ambiguous and the matrix elements can remain nonzero as the internuclear distance goes to infinity. We show that there is no ambiguity in the coupled equations, which describe inelastic collisions, and we demonstrate that the asymptotic couplings are required for a correct description of the asymptotic part of the scattering wave function. We develop on this basis a straightforward and simple procedure to deal with the asymptotic couplings in the framework of a full quantum approach and we apply it to a simple example of H+Na collisions.

Elimination of asymptotic couplings in atomic and molecular dynamics

Elimination of asymptotic couplings in atomic and molecular dynamics

Elimination of asymptotic couplings in atomic and molecular dynamics

The present work focuses on practical questions that inevitably arise on the topic of the elimination of asymptotic couplings in the treatment of electronic transitions in atomic and molecular dynamics. This includes the explicit form of the correction terms, the equivalence of ab initio approaches and their ease of implementation, as well as their indeterminacy and physical adequacy. Some related topics, such as Galilean invariance and high-v failure of molecular close-coupling expansions, are also discussed.

Erratum: Semiclassical multichannel perturbed-stationary-state model for rearrangement atomic collisions

We present a translationally invariant formulation of the semiclassical perturbed-stationary-state (PSS) method of atomic collisions that satisfies scattering boundary conditions without resorting to the electron translation factor. Our formulation hinges on the fact that correctly dissociating linear combinations of adiabatic electronic states become in the limit proper atomic states. Galilean invariant dynamical couplings are generated by scattering momenta conjugated to reaction coordinates in the Jacobi frames appropriate for describing either the colliding or the parting atomic species. Residual asymptotic couplings exist and constitute a necessary ingredient of our theory. They emerge because an electron in asymptotic capture (charge-exchange) state travels in the Jacobi frame proper for its collision entry state. As such, we do not eliminate the residual couplings by modifying the adiabatic functions with an electron translation factor, but rather harness them to construct asymptotic interaction-picture traveling states suitable for the PSS basis employed. This allows negating the traveling phases from the semiclassically propagated adiabatic amplitudes. The resulting phase-free capture amplitudes reach a definite asymptotic limit. The collision momentum operators in Jacobi coordinates proper for different asymptotic rearrangements are not equivalent. This is always true regardless of the basis employed, since imposing a diagonal internal kinetic energy in one Jacobi frame necessarily implies it is not diagonal in other internal coordinates (which is also the source of residual couplings). We therefore suggest a unique and instantaneous dynamical coupling operator may be constructed as the temporal adiabatic-state weighted average of the scattering momenta associated with the electronic rearrangements spanned by the basis. The proposed multichannel PSS propagator is shown, in forthcoming work, to faithfully reproduce the charge-exchange cross sections in collisions of ${\mathrm{He}}^{2+}$ on H and ${\mathrm{H}}^{+}$ on ${\mathrm{He}}^{+}$ up to the energies where ionization becomes important.

Semiclassical multichannel perturbed-stationary-state model for rearrangement atomic collisions.

We present a translationally invariant formulation of the semiclassical perturbed-stationary-state (PSS) method of atomic collisions that satisfies scattering boundary conditions without resorting to the electron translation factor. Our formulation hinges on the fact that correctly dissociating linear combinations of adiabatic electronic states become in the limit proper atomic states. Galilean invariant dynamical couplings are generated by scattering momenta conjugated to reaction coordinates in the Jacobi frames appropriate for describing either the colliding or the parting atomic species. Residual asymptotic couplings exist and constitute a necessary ingredient of our theory. They emerge because an electron in asymptotic capture (charge-exchange) state travels in the Jacobi frame proper for its collision entry state. As such, we do not eliminate the residual couplings by modifying the adiabatic functions with an electron translation factor, but rather harness them to construct asymptotic interaction-picture traveling states suitable for the PSS basis employed. This allows negating the traveling phases from the semiclassically propagated adiabatic amplitudes. The resulting phase-free capture amplitudes reach a definite asymptotic limit. The collision momentum operators in Jacobi coordinates proper for different asymptotic rearrangements are not equivalent. This is always true regardless of the basis employed, since imposing a diagonal internal kinetic energy in one Jacobi frame necessarily implies it is not diagonal in other internal coordinates (which is also the source of residual couplings). We therefore suggest a unique and instantaneous dynamical coupling operator may be constructed as the temporal adiabatic-state weighted average of the scattering momenta associated with the electronic rearrangements spanned by the basis. The proposed multichannel PSS propagator is shown, in forthcoming work, to faithfully reproduce the charge-exchange cross sections in collisions of ${\mathrm{He}}^{2+}$ on H and ${\mathrm{H}}^{+}$ on ${\mathrm{He}}^{+}$ up to the energies where ionization becomes important.

Elimination of asymptotic couplings in molecular dynamics

Two successful procedures that eliminate asymptotic couplings in close-coupling treatments of atomic dynamics are shown to be generalizable to systems with three or more atoms. The procedures are easily implemented, and employ either common generalized coordinates or common electron translation factors.

Evolution of molecular structure

Abstract In several areas of chemical physics, a knowledge of the evolution of the electronic structure of molecular systems along some path of nuclear coodinates is of interest. This paper points out the usefulness of a picture of evolution through wavefunction mixing. The degree of mixing can be gauged from the calculation of delayed overlap matrix elements. Taking directional derivatives as guideline, the paper presents some elementary patterns of evolution, involving narrow as well as wider pseudocrossings, conical intersections, overlap effects, polarization and hybridization, charge transfer and asymptotic couplings.

A solution to the infrared paradox

The infrared paradox is an inconsistency in the current derivation of the Einstein coefficient for vibrational excitations in molecular spectroscopy. A solution to the paradox is proposed by extending well established developments in the theory of atomic collisions to the case of vibrating molecules. When delay effects are unimportant, the new method does not require any change in the standard calculation of frequencies or intensities—only a reinterpretation of the nuclear wave functions. The improved formalism eliminates the problem of asymptotic couplings in molecular spectroscopy, as well as a basic difficulty in the usual picture of molecular structure.

Electron-translation effects in heavy-ion scattering

The origin and importance of electron-translation effects within a molecular description of electronic excitations in heavy-ion collisions is investigated. First, a fully consistent quantum-mechanical description of the scattering process is developed; the electrons are described by relativistic molecular orbitals, while the nuclear motion is approximated nonrelativistically. Leaving the quantum-mechanical level by using the semiclassical approximation for the nuclear motion, a set of coupled differential equations for the occupation amplitudes of the molecular orbitals is derived. In these coupled-channel equations the spurious asymptotic dynamical couplings are corrected for by additional matrix elements stemming from the electron translation. Hence, a molecular description of electronic excitations in heavy-ion scattering has been achieved, which is free from the spurious asymptotic couplings of the conventional perturbated stationary-state approach. The importance of electron-translation effects for continuum electrons and positrons is investigated. To this end an algorithm for the description of continuum electrons is proposed, which for the first time should allow for the calculation of angular distributions for $\ensuremath{\delta}$ electrons. Finally, the practical consequences of electron-translation effects are studied by calculating the corrected coupling matrix elements for the Pb-Cm system and comparing the corresponding $K$-vacancy probabilities with conventional calculations. We critically discuss conventional methods for cutting off the coupling matrix elements in coupled-channel calculations.