Ehrenfest trajectories

Ehrenfest trajectory for three-dimensional D + H2 generated by the RWP method, that is, the modified Hamiltonian operator f H). Dotted curves in (b) correspond to the Ehrenfest trajectory determined by the usual Schrddinger equation. See text for further details. 

Figure 2.27 (a) State Sj occupation probability as a function of time (b) energies of the and So states (strong coupling regions are indicated by circles) for an MMVB quasi-classical/Ehrenfest trajectory on azulene. (Adapted from Klein, S., Bearpark, M.J., Smith, B.R., Robb, M.A., Olivucci, M. and Bernardi, F., Chem. Phys. Lett, 292, 259-266, 1998.)... 

Shalashilin DV (2009) Quantum mechanics with the basis set guided by Ehrenfest trajectories theory and application to spin-boson model. J Chem Phys 130 244101... 

To add non-adiabatic effects to semiclassical methods, it is necessary to allow the trajectories to sample the different surfaces in a way that simulates the population transfer between electronic states. This sampling is most commonly done by using surface hopping techniques or Ehrenfest dynamics. Recent reviews of these methods are found in [30-32]. Gaussian wavepacket methods have also been extended to include non-adiabatic effects [33,34]. Of particular interest here is the spawning method of Martinez, Ben-Nun, and Levine [35,36], which has been used already in a number of direct dynamics studies. 

The center of the wavepacket thus evolves along the trajectory defined by classical mechanics. This is in fact a general result for wavepackets in a hannonic potential, and follows from the Ehrenfest theorem [147] [see Eqs. (154,155) in Appendix C]. The equations of motion are straightforward to integrate, with the exception of the width matrix, Eq. (44). This equation is numerically unstable, and has been found to cause problems in practical applications using Morse potentials [148]. As a result, Heller inboduced the P-Z method as an alternative propagation method [24]. In this, the matrix A, is rewritten as a product of matrices... 

The MMVB force field has also been used with Ehrenfest dynamics to propagate trajectories using mixed-state forces [84]. The motivation for this is... 

Quantum chemical methods, exemplified by CASSCF and other MCSCF methods, have now evolved to an extent where it is possible to routinely treat accurately the excited electronic states of molecules containing a number of atoms. Mixed nuclear dynamics, such as swarm of trajectory based surface hopping or Ehrenfest dynamics, or the Gaussian wavepacket based multiple spawning method, use an approximate representation of the nuclear wavepacket based on classical trajectories. They are thus able to use the infoiination from quantum chemistry calculations required for the propagation of the nuclei in the form of forces. These methods seem able to reproduce, at least qualitatively, the dynamics of non-adiabatic systems. Test calculations have now been run using duect dynamics, and these show that even a small number of trajectories is able to produce useful mechanistic infomiation about the photochemistry of a system. In some cases it is even possible to extract some quantitative information. 

Ehrenfest dynamics with the MMVB method has also been applied to the study of intermolecular energy transfer in anthryl-naphthylalkanes [85]. These molecules have a naphthalene joined to a anthracene by a short alkyl —(CH)n— chain. After exciting the naphthalene moiety, if n = 1 emission is seen from both parts of the system, if n = 3 emission is exclusively from the anthracene. The mechanism of this energy exchange is still not clear. This system is at the limits of the MMVB method, and the number of configurations required means that only a small number of trajectories can be run. The method is also unable to model the zwitterionic states that may be involved. Even so, the calculations provide some mechanistic information, which supports a stepwise exchange of energy, rather than the conventional direct process. 

The classical-path approximation introduced above is common to most MQC formulations and describes the reaction of the quantum DoF to the dynamics of the classical DoF. The back-reaction of the quantum DoF onto the dynamics of the classical DoF, on the other hand, may be described in different ways. In the mean-field trajectory (MFT) method (which is sometimes also called Ehrenfest model, self-consistent classical-path method, or semiclassical time-dependent self-consistent-field method) considered in this section, the classical force F = pj acting on the nuclear DoF xj is given as an average over the quantum DoF... 

Fig. 3.2. Two-dimensional potential energy surface V(R, 7) (dashed contours) for the photodissociation of C1CN, calculated by Waite and Dunlap (1986) the energies are given in eV. The closed contours represent the total dissociation wavefunction tot R,l E) defined in analogy to (2.70) in Section 2.5 for the vibrational problem. The energy in the excited state is Ef = 2.133 eV. The heavy arrow illustrates a classical trajectory starting at the maximum of the wavefunction and having the same total energy as in the quantum mechanical calculation. The remarkable coincidence of the trajectory with the center of the wavefunction elucidates Ehrenfest s theorem (Cohen-Tannoudji, Diu, and Laloe 1977 ch.III). Reprinted from Schinke (1990).

Nevertheless, this simple propagation method provides an intriguing picture of the evolution of the quantum mechanical wavepacket, at least for short times. It readily demonstrates that for short times the center of the wavepacket follows essentially a classical trajectory ( Ehrenfest s theorem, Cohen-Tannoudji, Diu, and Laloe 1977 ch.III). Figure 4.2 depicts an example the evolution of the two-dimensional wavepacket follows very closely the classical trajectory that starts initially with zero momenta at the Franck-Condon point. 

Classical trajectories are the backbones for the quantum mechanical wavefunctions ( Ehrenfest s theorem). If the dissociation is direct, a single trajectory, which starts near the equilibrium of the parent molecule, illustrates in a clear way the overall fragmentation mechanism. 

Gorban et al. in their works (Gorban, 2007 Gorban et al., 2001,2006) seems to be more comprehensive for our discussion. The works unfolded the idea of the Ehrenfests (1959) on the isolated system tending toward the Boltzmann equilibrium trajectory as a result of "agitations."... 

Applying Ehrenfest s theorem to the nuclear KS equation (68), the classical trajectory... 

Though it has been successfully applied, for example to describe energy transfer processes at metal surfaces, the Ehrenfest method fails when it becomes important to monitor different paths for different electronic states rather than a trajectory determined by an average over the different surfaces. This problem is particularly serious if one is interested in studying state specific nuclear pathways, such as those present in scattering events, or those determining low probability products in a chemical reaction. 

The methods outlined above can be contrasted with the Ehrenfest approach in which the bath follows a single mean field trajectory satisfying classical-like equations of motion R t) = P t)/M and P t) = Fmf = — ip t) dhei/dR ip t)). The mean field force on the trajectory is determined by the time dependent quantum subsystem state Cc t) a) which... 

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