Filtering schemes in the quantum-classical Liouville approach to nonadiabatic dynamics

Daniel A. Uken, Alessandro Sergi, and Francesco Petruccione

Phys. Rev. E 88, 033301 – Published 4 September 2013

Abstract

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.

Received 21 June 2013

DOI:https://doi.org/10.1103/PhysRevE.88.033301

Authors & Affiliations

Daniel A. Uken

School of Chemistry and Physics, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209 Pietermaritzburg, South Africa

Alessandro Sergi

School of Chemistry and Physics, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209 Pietermaritzburg, South Africa and National Institute for Theoretical Physics (NITheP), KwaZulu-Natal, South Africa

Francesco Petruccione

School of Chemistry and Physics and National Institute for Theoretical Physics, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban 4000, South Africa and National Institute for Theoretical Physics (NITheP), KwaZulu-Natal, South Africa

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Vol. 88, Iss. 3 — September 2013

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Figure 1
Comparison of the SSTP with the observable-cutting (filled square, panel a) and the transition-filtering algorithm (filled diamond, panel b) to path integral quantum results (open triangle) from Refs. [33, 34, 35]. System parameters were $β = 0.3$ , $ξ = 0.007$ , $Ω = \frac{1}{3}$ , corresponding to weak coupling. The value of the threshold parameter for the observable cutting was $κ_{t} = 100.0$ , and the value of the control parameter for the energy conserving filtering was $κ_{E} = 0.005$ . Two nonadiabatic transitions were included in the calculations.Reuse & Permissions
Figure 2
Comparison of the SSTP with the combined filtering algorithm (filled circle) to path integral quantum results (open triangle) from Refs. [33, 34, 35]. System parameters were $β = 0.3$ , $ξ = 0.007$ , $Ω = \frac{1}{3}$ . The value of the threshold parameter was $κ_{t} = 1.5$ , and the value of the energy conserving filtering control parameter was $κ_{E} = 0.005$ . Two non-adiabatic transitions were included in the calculations.Reuse & Permissions
Figure 3
Comparison of the SSTP with the observable-cutting (filled square, panel a) and the transition-filtering algorithm (filled diamond, panel b) to path integral quantum results (open triangle) from Ref. [36]. System parameters were $β = 12.5$ , $ξ = 0.09$ , $Ω = 0.4$ , corresponding to mid-range coupling. The value of the threshold parameter for the direct filtering was $κ_{t} = 50.0$ , and the value of the control parameter was $κ_{E} = 0.025$ . Two nonadiabatic transitions were included in the calculations.Reuse & Permissions
Figure 4
Comparison of the SSTP with the combined filtering algorithm (filled circle) to path integral quantum results (open triangle) from Ref. [36]. System parameters were $β = 12.5$ , $ξ = 0.09$ , $Ω = 0.4$ . The value of the threshold parameter was $κ_{t} = 3.5$ , and the value of the control parameter was $κ_{E} = 0.05$ . Two nonadiabatic transitions were included in the calculations.Reuse & Permissions
Figure 5
Comparison of the SSTP with the observable-cutting (filled square, panel a) and the transition-filtering algorithm (filled diamond, panel b) to path integral quantum results (open triangle) from Ref. [38]. System parameters were $β = 0.25$ , $ξ = 2.0$ , $Ω = 1.2$ , corresponding to high coupling. The value of the bound parameter for the direct filtering was $κ_{t} = 50.0$ , and the value of the control parameter was $κ_{E} = 0.5$ . Two nonadiabatic transitions were included in the calculations.Reuse & Permissions
Figure 6
Comparison of the SSTP with the combined filtering algorithm (filled circle) to path integral quantum results (open triangle) from Ref. [38]. System parameters were $β = 0.25$ , $ξ = 2.0$ , $Ω = 1.2$ . The value of the threshold parameter was $κ_{t} = 5.0$ , and the value of the control parameter was $κ_{E} = 1.0$ . Two nonadiabatic transitions were included in the calculations.Reuse & Permissions

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