Approximation momentum-jump

Within the momentum-jump approximation (58), the algorithm provides an in principle exact stochastic realization of the QCL equation. Apart from practical problems to be discussed below, the trajectory implementation therefore represents a well-defined computational scheme. 

In order to get a first impression on the performance of the QC Liouville approach, it is instructive to start with a simple one-mode spin-boson model, that is. Model IVa [205]. In what follows, the QCL calculations used the first-order Trotter scheme (61) with a time step 8r = 0.05 fs. If not noted otherwise, we have employed the momentum-jump approximation (59) and the initial number of random walkers employed was N = 50 000. 

The presence of the momentum derivatives in J makes the action of this operator difficult to simulate, because it acts on all functions to its right. This will generate a branching tree of trajectories. This difficulty is avoided by making the momentum-jump approximation. To see how this approximation is obtained, the following change of variables is made ... 

The approximations surrounding the definition of the J operator comprise the momentum-jump approximation. This translation or shift of the momentum corresponds precisely to the amount of energy transferred during a transition... 

When the quantum-classical Liouville equation is expressed in the adiabatic basis, the most difficult terms to simulate come from the off-diagonal force matrix elements, which give rise to the nonadiabatic coupling matrix elements. As described above, contributions coming from this term were computed using the momentum-jump approximation in the context of a surface-hopping scheme. 

The propagator e JS)ss, is responsible for quantum transitions and bath momentum changes. In order to compute its action, we use the momentum-jump approximation [12, 23] that replaces the small continuous momentum changes with momentum jumps that accompany each quantum transition. In this approximation, the matrix elements of e can be written in terms of a matrix M to 0(S2),... 

As noted earlier, the fundamental equations of the QCL dynamics approach are exact for this model, however, in order to implement these equations in the approach detailed in section 2 the momentum jump approximation of Eq.(14) is made in addition to the Trotter factorization of Eq.(12). Both approximations become more accurate as the size of the time step 5 is reduced. Consequently, the results presented below primarily serve as tests of the validity and utility of the momentum-jump approximation. For a discussion of other simulation schemes for QCL dynamics see Ref. [21] in this volume. The linearized approximate propagator is not exact for the spin-boson model. However when used as a short time approximation for iteration as outlined in section 3 the approach can be made accurate with a sufficient number of iterations [37]. 

Here we show the steps leading to this momentum-jump approximation . Since S / = AEai3dai3 with = Ea — we may write... 

If the transition is accepted, then, using the momentum jump approximation, we translate the momentum PAt to Pat = Pat + AP where AP is defined in (84). We then write... 

The approximation (58) resembles the usual momentum-jump ansatz employed in various surface-hopping methods [55, 57, 58, 61, 66, 67, 82]. In order to determine the momentum shift of a trajectory, however, the latter formulations typically require the conservation of nuclear energy ... 

In the first line of this equation we made the main assumption that the first two terms on the left hand side could be approximated by the exponential of the operator. In the second line we wrote the momentum vector as a sum of its components along dap and perpendicular to 4/3) nd in the penultimate line we used the fact that the exponential operator is a translation operator in the variable (P dap). In the last line the momentum jump AP is given by... 

It is noted that the momentum derivatives of the coupling elements (51) represent one of the main obstacles of a practical trajectory-based evaluation of the QCL equation, because these terms require the knowledge of the function in question not only at a particular point in phase space but at the same time also at nearby points. As a remedy, we may restrict ourselves to the limit of small momentum changes Snm/P At 1 and approximate 1 - - SnmP exp(S m5/dP). Since eP / Pf p) = /(p + S ), the approximation reduces the action of the differential operator to a simple shift of momenta. We note that this approximation resembles the usual momentum-jump ansatz employed in various surface-hopping methods. ... 

In the case of a hydraulic jump on a sloping channel, it is simply necessary to add the sine component of the weight of the water to the general momentum equation. Although there exist more refined methods for doing this [41], a good approximation may be obtained by assuming the jump section to be a trapezoid with bases jq and y2 and altitude of about 6Y2. 

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