Adiabatic mapping method

With the advent of molecular dynamics simulations applied to carbohydrates, one can anticipate the direct computation of more conceptually appealmg surfaces of V in 0s) space from a given U( qint,qext)) in the near future. Monte Carlo integration over (qext) and (b,x, 0h) for fix (0s) provides an alternative procedure, but one which is probably less attractive in terms of efficiency than the molecular dynamics approach. A second alternative, known as adiabatic mapping, provides an approximation to V((0s ), and applications of this method to carbohydrates have recently begun to appear. 12,13 in this approach the conformational... 

Figure 19. Time-dependent (a) diabatic and (b) adiabatic electronic excited-state populations and (c) vibrational mean positions as obtained for Model 1. Shown are results of the mean-field trajectory method (dotted lines), the quasi-classical mapping approach (thin full lines), and exact quantum calculations (thick full lines).

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. 

In recent work [16, 17] we presented a new mixed quantum-classical method, which we call LAND-Map (Linearized approach to non-adiabatic dynamics in the mapping formulation), for calculating correlation functions. The method couples the linearization ideas put forth by various workers [18-26] with the mapping description of non-adiabatic transitions [27-31]. 

In order to extend the linearization scheme to non-adiabatic dynamics it is convenient to represent the role of the discrete electronic states in terms of operators that simplify the evolution of the quantum subsystem with out changing its effect on the classical bath. A way to do this was first suggested by Miller, McCurdy and Meyer [28,29[ and has more recently been revisited by Thoss and Stock [30, 31[. Their method, known as the mapping formalism, represents the electronic degrees of freedom and the transitions between different states in terms of positions and momenta of a set of fictitious harmonic oscillators. Formally the approach is exact, but approximations (e.g. semi-classical, linearized SC-IVR, etc.) must be made for its numerical implementation. 

S. Bonella and D. F. Coker (2001) Semi-classical implementation of mapping Hamiltonian methods for general non-adiabatic problems. Chem,. Phys. 268, p. 323... 

Finally, we consider Model III, which describes an ultrafast photoin-duced isomerization process. Figure 13 shows quantum mechanical results as well as results of the ZPE-corrected mapping approach for three different observables the adiabatic and diabatic population of the excited state, and Pcis> the probability that the system remains in the initially prepared cis-conformation. A ZPE correction of 7 = 0.5 has been used in the mapping calculation, based on the criterion to reproduce the quantum mechanical long-time limit of the adiabatic population. It is seen that the ZPE-corrected mapping approach represents an improvement compared to the mean-field trajectory method (cf. Fig. 3), in particular for the adiabatic population. The influence of the ZPE correction on the dynamics of the observables is, however, not as large as in the two other models. 

The latter method is based on a least-squares fitting of the numerical map of the adiabatic potential to the analytic and is applicable in an arbitrary degree of com-... 

Several algorithms have been constructed to simulate the solution of the QCLE. The simulation methods usually utilize particular representations of the quantum subsystem. Surface-hopping schemes that make use of the adiabatic basis have been constructed density matrix evolution has been carried out in the diabatic basis using trajectory-based methods, some of which make use of a mapping representation of the diabatic states.A representation of the dynamics in the force basis has been implemented to simulate the dynamics using the multithreads algorithm. ... 

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