Hopping trajectories

The prefactor for the HK propagator contains the derivatives 3q/3Fo, 0Ft/3po, 3pt/3Fo, and 3pt/3po. These derivative are evaluated for hopping trajectories, as discussed above, such that the changes in the hopping points, accompanying changes in the initial phase space point for the trajectory, always occur in a direction... 

The surface-hopping trajectories obtained in the adiabatic representation of the QCLE contain nonadiabatic transitions between potential surfaces including both single adiabatic potential surfaces and the mean of two adiabatic surfaces. This picture is qualitatively different from surface-hopping schemes [2,56] which make the ansatz that classical coordinates follow some trajectory, R(t), while the quantum subsystem wave function, expanded in the adiabatic basis, is evolved according to the time dependent Schrodinger equation. The potential surfaces that the classical trajectories evolve along correspond to one of the adiabatic surfaces used in the expansion of the subsystem wavefunction, while the subsystem evolution is carried out coherently and may develop into linear combinations of these states. In such schemes, the environment does not experience the force associated with the true quantum state of the subsystem and decoherence by the environment is not automatically taken into account. Nonetheless, these methods have provided com-... 

The chapter is organized as follows The quantum-classical Liouville dynamics scheme is first outlined and a rigorous surface hopping trajectory algorithm for its implementation is presented. The iterative linearized density matrix propagation approach is then described and an approach for its implementation is presented. In the Model Simulations section the comparable performance of the two methods is documented for the generalized spin-boson model and numerical convergence issues are mentioned. In the Conclusions we review the perspectives of this study. 

While the two methods are, at face value, quite different in the ways in which full quantum dynamics is reduced to quantum-classical dynamics, there are common elements in the manner in which they are simulated. The Trotter-based scheme for QCL dynamics makes use of the adiabatic basis and is based on surface-hopping trajectories where transitions are sampled by a Monte Carlo scheme that requires reweighting. Similarly, ILDM calculations make use of the mapping hamiltonian basis and also involve a similar Monte Carlo sampling with reweighting of trajectories in the ensemble used to obtain the expectation values of quantum operators. 

One should note that the MEPs shown are not true dynamical paths, which of course can only be obtained by dynamical calculations. We have carried these out [6,9] using several different dynamical descriptions, including surface hopping trajectories [95,96]. The resulting dynamical path for the slow solvent is reasonably similar to the MEP, but this is not the case for the fast solvent, a point to which we return below. A further dynamical study [6] has compared, for the fast solvent case using surface hopping trajectories, the dynamics with the present nonequilibrium solvation description to those when equilibrium solvation is assumed. This is the most favorable case for the validity... 

Nieber and Doltsinis [64] calculated 15 nonadiabatic surface hopping trajectories starting from configurations and velocities sampled from a ground state simulation. The nonradiative excited state lifetime has been determined by fitting the time-dependent, decaying excited state ensemble population to a mono-exponential function subject to the boundary condition that all molecules were in the Sj state... 

An ensemble of 16 nonadiabatic surface hopping trajectories have been calculated sampling different initial conditions from a 300 K ground state simulation. A monoexponential fit to the Sj population gives a lifetime of 1.0 ps. Estimating the lifetime using the relation (10-13) yields the interval [0.9...1.6. ..6.0] ps, which indicates that the result from the exponential fit probably underestimates the lifetime [41,42], The 7H-keto tautomer is thus considerably longer lived than the 9H-keto form. 

Langer and Doltsinis [45] have calculated nonadiabatic surface hopping trajectories for 10 different initial configurations sampled from a ground state AIMD runs at 100 K. They later extended their study to a total of 16 trajectories [41, 42], From a mono-exponential fit to the 5) population a lifetime of 1.3 ps is obtained (see Table 10-1 the average transition probability and its standard deviation leads to the interval [0.6...1.1...3.5] ps. Thus methylation appears to result in a slightly longer excited state lifetime. 

H-keto G in liquid water A ground state simulation of 9H-keto G embedded in 60 H20 molecules in a periodic setup at 300 K has been performed from which six configurations have been randomly selected as input for 6 nonadiabatic surface hopping trajectory calculations starting in the S1 excited state. Comparison with the simulations in the gas phase (see Section 10.3.3.2.1) permits analysis of the effects of the water solvent on the mechanism of radiationless decay. 

Figure 10-20. Comparison of the energy gap AE grey lines) and the Hh >N<3) distance fi(NH) top panel, black line), the C O NWcW dihedral angle middle panel, black line), and the H 4b N 4 1 C 4 1 C 3 dihedral angle bottom panel, black line) for a typical surface hopping trajectory of GC...

On the basis of the ensemble of 10 surface hopping trajectories an excited state lifetime in aqueous solution ofr = 115 9fs has been estimated assuming monoexponential decay [55], Fitting a bi-exponential function yields a fast component of Tj = 29 9 fs and a slow component of r2 = 268 62 fs [55], These results are very similar to the gas phase values. However, the latter should be considered to be much more accurate due to the much larger number of trajectories. 

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