Exact Factorization of the Electron-Nuclear Wavefunction: Fundamentals and Algorithms

Investigating the Photodynamics of trans-Azobenzene with Coupled Trajectories

“…The fourth-order Runge–Kutta (RK4) integrator was employed for the propagation of the electronic coefficients since the presence of the term Ċ m , c t α in eq prevents us to use the propagator employed in the LD scheme (see Appendix). As discussed previously in the literature, ,, the exact factorization allows us to formulate the coupled dynamics of electrons and nuclei in the vicinity of conical interactions without invoking the very concept of conical intersections or of geometric phases. Nonetheless, when an algorithm is derived, CT-TSH in this case, which uses the adiabatic electronic representation in order to employ standard quantum-chemistry methods for electronic structure, geometric-phase effects (as well as diagonal BO corrections that compensate the effect of the geometric phase) should be accounted for, and this is not done in the general derivation of CT-TSH …”

“…Quantum decoherence is the main example, as well as quantum interference. ,− In this respect, the quantum-classical numerical approaches derived from the exact factorization seem to show an improvement in the description of quantum decoherence, compared to, for example, the Tully surface hopping (TSH) method (one of the most widely used quantum-classical methods in the community) . Exact-factorization-based approaches allow to recover some quantum effects, thanks to the coupling among the trajectories. − In particular, the recently developed coupled-trajectory Tully surface hopping (CT-TSH) method combines the basic idea of a surface-hopping procedure, i.e., classical adiabatic nuclear evolution using stochastic hops, with the “decoherence-corrected” electronic quantum dynamics, where the additional term accounting for decoherence effects is rigorously derived from the exact factorization . (In some sense, then, CT-TSH should not be viewed as another decoherence scheme for surface hopping.…”

Exact-factorization study of the photochemistry of phenol

“…Specifically, in both equations, the first two terms are identical to the Ehrenfest scheme. The additional contributions allow one to describe decoherence effects, as it has been extensively discussed in the literature 37,48,64 . Furthermore, the extra term in the electronic evolution equation ( 14) provides a new mechanism for population transfer mediated by the quantum momentum, that has been shown to be key in accurately capturing multi-state processes such as dynamics through a three-state conical intersection 35,53 .…”

“…Ciccotti and coworkers rigorously reformulated 42,43 the nuclear equation using the method of characteristics, that can be interpreted as quantum trajectories in a Bohmian sense 44 , but the main computational advantage has been achieved with a more classical-like treatment of the nuclear dynamics 33,[45][46][47] . Algorithms such as the coupled-trajectory mixed quantum-classical (CTMQC) scheme 45,[48][49][50] , along with its energyconserving variation CTMQC-E 51 , and coupled-trajectory Tully surface hopping (CTTSH) 47 allowed us to simulate nonadiabatic processes of various molecular models 36,51,52 and molecular systems using an on-the-fly approach 30,38,53 .…”

Adiabatic and Nonadiabatic Dynamics with Interacting Quantum Trajectories

“…The numerical calculation of the TDPES has been carried out as follows: (1) exact vibronic wavepacket dynamics calculations have been performed with the ElVibRot program 54 on the Tully models; (2) the output of ElVibRot has been analyzed by the Exact Factorization Analysis Code (EFAC) 55 to produce the TDPES. The TDPES can be easily expressed in terms of "standard" (a)diabatic nuclear amplitudes (and their spatial and time derivatives) arising from the Born-Huang representation of the molecular wavefunction.…”