Direct dynamics semiclassical

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. 

Direct dynamics attempts to break this bottleneck in the study of MD, retaining the accuracy of the full electronic PES without the need for an analytic fit of data. The first studies in this field used semiclassical methods with semiempirical [66,67] or simple Hartree-Fock [68] wave functions to heat the electrons. These first studies used what is called BO dynamics, evaluating the PES at each step from the elech onic wave function obtained by solution of the electronic structure problem. An alternative, the Ehrenfest dynamics method, is to propagate the electronic wave function at the same time as the nuclei. Although early direct dynamics studies using this method [69-71] restricted themselves to adiabatic problems, the method can incorporate non-adiabatic effects directly in the electionic wave function. 

Full quantum wavepacket studies on large molecules are impossible. This is not only due to the scaling of the method (exponential with the number of degrees of freedom), but also due to the difficulties of obtaining accurate functions of the coupled PES, which are required as analytic functions. Direct dynamics studies of photochemical systems bypass this latter problem by calculating the PES on-the-fly as it is required, and only where it is required. This is an exciting new field, which requires a synthesis of two existing branches of theoretical chemistry—electronic structure theory (quantum chemistiy) and mixed nuclear dynamics methods (quantum-semiclassical). 

For larger systems, various approximate schemes have been developed, called mixed methods as they treat parts of the system using different levels of theory. Of interest to us here are quantum-semiclassical methods, which use full quantum mechanics to treat the electrons, but use approximations based on trajectories in a classical phase space to describe the nuclear motion. The prefix quantum may be dropped, and we will talk of semiclassical methods. There are a number of different approaches, but here we shall concentrate on the few that are suitable for direct dynamics molecular simulations. An overview of other methods is given in the introduction of [21]. 

In order to investigate which model for the behavior of 8 is closer to being correct, our group provided Carpenter with an analytical expression, fit to our PES, so that Carpenter could perform semiclassical trajectory calculations on our PES. At the same time. Doubleday and Hase undertook direct dynamics calculations. As discussed in Chapter 21 in this volume, in the latter type of trajectory calculation the forces acting on a molecule at different points on a PES are found by performing electronic structure calculations. For this purpose. Doubleday and Hase used a reparameterized version of AMI that provided a PES, similar to those calculated by Getty and by Baldwin, Yamaguchi, and Schaefer. 

Combining all of these ideas of this section gives rise to Bom-Oppenheimer MD, sometimes also referred to as semiclassical dynamics. Separation of electronic and nuclear motion is assumed, namely, the Bom-Oppenheimer approximation. Atoms move on the electronic PES, computed using some QM method, following the classical equations of motion. In summary, the steps for computing trajectories using direct dynamics are the following. 

Ohrn and co-workers have developed a direct dynamics approach which incorporates both the electrons and nuclei dynamics (END).""" The complete electron-nuclear coupling terms are retained in the calculation and, as a result, the dynamics is not constrained to a single Born-Oppenheimer potential energy surface i.e., electronic non-adiabaticity is explicitly included. A complication in this approach is the computational demand in propagating an electronic wavefunction which is an accurate representation of the ground electronic state as well as multiple excited electronic states. This approach will become more widely used as computation becomes more powerful. In its initial development,""" Deumens et al. used END and treated the dynamics of the nuclei purely classical as in the above classical direct dynamics. More recently, a semiclassical description of the nuclear motion has been implemented by incorporating Heller s""" "" Gaussian wave packet dynamics."" ... 

For example, the ZN theory, which overcomes all the defects of the Landau-Zener-Stueckelberg theory, can be incorporated into various simulation methods in order to clarify the mechanisms of dynamics in realistic molecular systems. Since the nonadiabatic coupling is a vector and thus we can always determine the relevant one-dimensional (ID) direction of the transition in multidimensional space, the 1D ZN theory can be usefully utilized. Furthermore, the comprehension of reaction mechanisms can be deepened, since the formulas are given in simple analytical expressions. Since it is not feasible to treat realistic large systems fully quantum mechanically, it would be appropriate to incorporate the ZN theory into some kind of semiclassical methods. The promising semiclassical methods are (1) the initial value... 

The widths and lifetimes obtained are summarized in Fig. 10 for different initial states (fc=l,2,3,6 and 12). The large difference between the hfetimes between A and B electronic states persists. Also, the lifetimes obtained for A state here are of 1-10 ps, while the semiclassical estimates obtained previously[5 8] are of 0.01-0.1 ps. This difference can be attributed to the difference on the dynamical method used and to the fact that the non-adiabatic couplings were not directly calculated in this last work[59]. 

An equivalent treatment was proposed by Lambert [176] to account for the field-induced intensity changes in adsorbed CO in UHV and at electrodes. The proposed semiclassical equation is based on the fact that the integrated intensity is proportional to the dynamic dipole moment 0///00 (or (dM /dOf) for the component in the direction of the applied field. The deduced expression for the change in intensity with the applied field is ... 

This paper reviews this classical S-matrix theory, i.e. the semiclassical theory of inelastic and reactive scattering which combines exact classical mechanics (i.e. numerically computed trajectories) with the quantum principle of superposition. It is always possible, and in some applications may even be desirable, to apply the basic semiclassical model with approximate dynamics Cross7 has discussed the simplifications that result in classical S-matrix theory if one treats the dynamics within the sudden approximation, for example, and shown how this relates to some of his earlier work8 on inelastic scattering. For the most part, however, this review will emphasize the use of exact classical dynamics and avoid discussion of various dynamical models and approximations, the reason being to focus on the nature and validity of the basic semiclassical idea itself, i.e., classical dynamics plus quantum superposition. Actually, all quantum effects—being a direct result of the superposition of probability amplitudes—are contained (at least qualitatively) within the semiclassical model, and the primary question to be answered regards the quantitative accuracy of the description. 

The basic semiclassical idea is that one uses a quantum mechanical description of the process of interest but then invokes classical mechanics to determine all dynamical relationships. A transition from initial state i to final state f, for example, is thus described by a transition amplitude, or S-matrix element Sfi, the square modulus of which is the transition probability Pf = Sfi 2. The semiclassical approach uses classical mechanics to construct the classical-limit approximation for the transition amplitude, i.e. the classical S-matrix the fact that classical mechanics is used to construct an amplitude means that the quantum principle of superposition is incorporated in the description, and this is the only element of quantum mechanics in the model. The completely classical approach would be to use classical mechanics to construct the transition probability directly, never alluding to an amplitude. 

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