Efficient discretization of the continuum through complex contour deformation

Shenvi, Neil; Schmidt, Jonathan; Edwards, Stephen T.; Tully, John C.

doi:10.1103/physreva.78.022502

Cited by 24 publications

(24 citation statements)

References 20 publications

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“…More generally, near a metal surface, Shenvi et al have suggested discretizing the electronic bath, and running SH on a large number of independent potential energy surfaces (PESs). [18][19][20] A nonequilibrium version of the Shenvi algorithm might be possible in the vein of Refs. 21-26. In a previous paper, 27 which we refer to as Paper II, we have analyzed an alternative approach based on a classical master equation (CME) that describes the dynamics of the AH impurity subsystem.…”

Section: Introductionmentioning

confidence: 99%

Surface hopping with a manifold of electronic states. III. Transients, broadening, and the Marcus picture

Dou

Nitzan

Subotnik

2015

The Journal of Chemical Physics

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In a previous paper [Dou et al., J. Chem. Phys. 142, 084110 (2015)], we have introduced a surface hopping (SH) approach to deal with the Anderson-Holstein model. Here, we address some interesting aspects that have not been discussed previously, including transient phenomena and extensions to arbitrary impurity-bath couplings. In particular, in this paper we show that the SH approach captures phonon coherence beyond the secular approximation, and that SH rates agree with Marcus theory at steady state. Finally, we show that, in cases where the electronic tunneling rate depends on nuclear position, a straightforward use of Marcus theory rates yields a useful starting point for capturing level broadening. For a simple such model, we find I-V curves that exhibit negative differential resistance.

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Section: Introductionmentioning

confidence: 99%

Surface hopping with a manifold of electronic states. III. Transients, broadening, and the Marcus picture

Dou

Nitzan

Subotnik

2015

The Journal of Chemical Physics

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“…Direct discretization strategies Let us consider the approach of Ref. 25 and rephrase the problem of discretizing the Hamiltonian as that of discretizing the integral in (5). The simplest approximation for an integral is obtained by using a trapezoidal integration rule…”

Section: Relation Of Different Discretization Strategiesmentioning

confidence: 99%

How to discretize a quantum bath for real-time evolution

2015

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Many numerical techniques for the description of quantum systems that are coupled to a continuous bath require the discretization of the latter. To this end, a wealth of methods has been developed in the literature, which we classify as (i) direct discretization, (ii) orthogonal polynomial, and (iii) numerical optimization strategies. We recapitulate strategies (i) and (ii) to clarify their relation. For quadratic Hamiltonians, we show that (ii) is the best strategy in the sense that it gives the numerically exact time evolution up to a maximum time tmax, for which we give a simple expression. For non-quadratic Hamiltonians, we show that no such best strategy exists. We present numerical examples relevant to open quantum systems and strongly correlated systems, as treated by dynamical mean-field theory (DMFT).

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“…In general, however, the FSSH algorithm has usually been restricted to studying isolated molecules in solvents with only a handful of electronic states. 38,68 Thus far, the most important exception to this general rule was the pioneering "Independent Electron Surface Hopping (IESH)" model of Shenvi, Roy, and Tully, [39][40][41] who studied NO scattering off of a gold surface. Shenvi et al suggested discretizing a continuum of adiabatic electronic levels to simulate electronic friction in a metal; by running FSSH on a large number of electronic states, Shenvi et al were able to correctly describe vibrational relaxation of the NO molecule.…”

Section: Introductionmentioning

confidence: 99%

Surface hopping with a manifold of electronic states. II. Application to the many-body Anderson-Holstein model

Dou

Nitzan

Subotnik

2015

The Journal of Chemical Physics

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We investigate a simple surface hopping (SH) approach for modeling a single impurity level coupled to a single phonon and an electronic (metal) bath (i.e., the Anderson-Holstein model). The phonon degree of freedom is treated classically with motion along--and hops between--diabatic potential energy surfaces. The hopping rate is determined by the dynamics of the electronic bath (which are treated implicitly). For the case of one electronic bath, in the limit of small coupling to the bath, SH recovers phonon relaxation to thermal equilibrium and yields the correct impurity electron population (as compared with numerical renormalization group). For the case of out of equilibrium dynamics, SH current-voltage (I-V) curve is compared with the quantum master equation (QME) over a range of parameters, spanning the quantum region to the classical region. In the limit of large temperature, SH and QME agree. Furthermore, we can show that, in the limit of low temperature, the QME agrees with real-time path integral calculations. As such, the simple procedure described here should be useful in many other contexts.

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Efficient discretization of the continuum through complex contour deformation

Cited by 24 publications

References 20 publications

Surface hopping with a manifold of electronic states. III. Transients, broadening, and the Marcus picture

Surface hopping with a manifold of electronic states. III. Transients, broadening, and the Marcus picture

How to discretize a quantum bath for real-time evolution

Surface hopping with a manifold of electronic states. II. Application to the many-body Anderson-Holstein model

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