Trajectory surface hopping nonadiabatic transition

In order to test the small x assumptions in our calculations of condensed phase vibrational transition probabilities and rates, we have performed model calculations, - for a colinear system with one molecule moving between two solvent particles. The positions ofthe solvent particles are held fixed. The center of mass position of the solute molecule is the only slow variable coordinate in the system. This allows for the comparison of surface hopping calculations based on small X approximations with calculations without these approximations. In the model calculations discussed here, and in the calculations from many particle simulations reported in Table II, the approximations made for each trajectory are that the nonadiabatic coupling is constant that the slopes of the initial and final... 

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-... 

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. 

Figure 10-14. Time evolution of the nonadiabatic surface hopping parameter, P10 (Eq. 10-10), for a transition from the 5 excited state to the. V0 ground state for representative 7Me-keto (fast oscillating, small amplitude dark grey curve) and 9Me-keto (fast oscillating, large amplitude light grey curve) G trajectories. The steep increase of P10 at / 10 fs in the case of 9Me-keto coincides with the transition from a quasi-planar to an out-of-plane distorted structure. At / 40 fs the amino group starts rotating...

Figure 10-21. Comparison of the nonadiabatic transition parameter, P]n (see Eq. 10-10, grey lines), and the time-derivatives (black lines) of the energy gap AEifirst panel), the Hd N -1) distance (secondpanel), the C O N C 41 dihedral angle (thirdpanel), and the H 4blN 4lC 4lC(51 dihedral angle (fourthpanel) for a typical surface hopping trajectory of GC...

Yet, some theoretical problems are left to be discussed to seek for the ultimate and idealistic features as a nonadiabatic-transition theory Although a trajectory thus hopping plural times converges to run on an adiabatic potential surface asymptotically, the off-diagonal density matrix element Pij t) does not vanish practically, as in the original SET. This is ascribed to an incomplete treatment of the nuclear-electronic entanglement. This issue, often referred to as the problem of decoherence, is originated from the nuclear wavepacket bifurcation due to different slopes of potential surfaces, which will be discussed more precisely below. 

The plateau value of the time-dependent rate coefficient k t) = kAB t) + kBA t), which is the sum of the forward and reverse rate constants, determines the overall chemical relaxation time, tchem = for the proton transfer. The nonadiabatic time-dependent rate coefficient k t) is shown in Figure 10.2. The rate constant extracted from the plateau value of this plot is k = 0.163 ps Up to six nonadiabatic transitions were required to obtain converged results. An examination of the trajectories in the ensemble revealed that the major nonadiabatic correction to the rate comes from two quantum transitions ground state -> coherent state ground state. This picture of how nonadiabatic transitions influence the reaction rate is quite different from that in standard surface-hopping methods. 

A further complexity in the simulation of photochemistry and more in general of excited state photoinduced dynamics is that they are intrinsically nonadiabatic processes, in which the coupling between the nuclear and electronic motion leads to nonradialive transitions between electronic states. A generally applicable approach for this purpose is the mixed quantum-classical dynamics in which the nuclear motion is described by classical trajectories obtained in the framework of molecular dynamics on the fly combined with Tully s surface hopping (TSH) procedure... 

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