Quantum/semiclassical approaches approximation

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]. 

On the contrary, the semiclassical approach in the problem of the optical absorption is restricted to a great extent and the adequate description of the phonon-assisted optical bands with a complicated structure caused by the dynamic JTE cannot be done in the framework of this approach [13]. An expressive example is represented by the two-humped absorption band of A —> E <8> e transition. The dip of absorption curve for A —> E <8> e transition to zero has no physical meaning because of the invalidity of the semiclassical approximation for this spectral range due to essentially quantum nature of the density of the vibronic states in the conical intersection of the adiabatic surface. This result is peculiar for the resonance (optical) phenomena in JT systems full discussion of the condition of the applicability of the adiabatic approximation is given in Ref. [13]. 

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. 

This classical model has been extended further by Nakamura > and Miller by including interference effects between the incoming and outgoing portions of the trajectory this resulting semiclassical approach is equivalent to a WKB analysis of the quantum mechanical treatment, employing the stationary-phase approximation. ... 

From the chemically point of view, the valence states are those situated in the chemical zone -and they are the main concern forthe chemical reactivity by employing the frontier or the outer electrons consequently, the semiclas-sical approximation that models the excited states was expressly presented either as an extension of the quantum Feynman path integral or as a specialization of the Feynman-Kleinert formalism for higher temperature treatment of quantum systems (see Section 2.5). However, due to the correspondences of Table 2.1 one may systematically characterize the semiclassical (or quantum chemical) approaches as one of the limiting situations (Putz, 2009) ... 

The quantities lnd>(r (r)) are defined as the quotients (-Sp,lh), where is the so-called action for the problem under consideration and involves an integration of kinetic and potential contributions over the period 0dimensionless quantity - In (r" (t)), its relation to the product of the density matrix elements in Eqs. (14) and (16) being clear [28]. A few simple examples (e.g., free particle and harmonic oscillator) admit the exact application of the PI formahsm in the P t form [12, 13], but for general many-body quantum systems this is not possible. However, some analytic developments related to Eq. (15) have given rise to the so-called Feynman s semiclassical approaches, which will be considered in Section 111. To exploit the power of the PI formahsm computational schemes utilize finite-P discretizations. In this regard, given that approximations to calculate density matrix... 

For continuous pair potentials v(r) a number of useful semiclassical approaches can be derived from the PI formalism. They give rise to quantum effective pan-potentials, which are perfectly suited for quantum simulation studies of fluids and solids under a wide range of conditions (quantum exchange is excluded). All of them are interconnected either by limiting procedures (e.g., high temperatures) or by making truncation approximations up to certain orders of There are... 

Due to the complexity of a full quantum mechanical treatment of electron impact ionization, or even a partial wave approximation, for all but relatively simple systems, a large number of semiempirical and semiclassical formulae have been developed. These often make basic assumptions which can limit their range of validity to fairly small classes of atomic or molecular systems. The more successful approaches apply to broad classes of systems and can be very useful for generating cross sections in the absence of good experimental results. The success of such calculations to reproduce experimentally determined cross sections can also give insight into the validity of the approximations and assumptions on which the methods are based. 

In a recent analysis carried out for a bounded open system with a classically chaotic Hamiltonian, it has been argued that the weak form of the QCT is achieved by two parallel processes (B. Greenbaum et.al., ), explaining earlier numerical results (S. Habib et.al., 1998). First, the semiclassical approximation for quantum dynamics, which breaks down for classically chaotic systems due to overwhelming nonlocal interference, is recovered as the environmental interaction filters these effects. Second, the environmental noise restricts the foliation of the unstable manifold (the set of points which approach a hyperbolic point in reverse time) allowing the semiclassical wavefunction to track this modified classical geometry. 

For this reason, we will restrict our subsequent approach to planar configurations of the two electrons and of the nucleus, with the polarization axis within this plane. This presents the most accurate quantum treatment of the driven three body Coulomb problem to date, valid in the entire nonrelativistic parameter range, without any adjustable parameter, and with no further approximation beyond the confinement of the accessible configuration space to two dimensions. Whilst this latter approximation certainly does restrict the generality of our model, semiclassical scaling arguments suggest that the unperturbed three... 

The mapping approach outlined above has been designed to furnish a well-defined classical limit of nonadiabatic quantum dynamics. The formalism applies in the same way at the quantum-mechanical, semiclassical (see Section VIII), and quasiclassical level, respectively. Most important, no additional assumptions but the standard semiclassical and quasi-classical approximations are needed to get from one level to another. Most of the established mixed quantum-classical methods such as the mean-field-trajectory method or the surface-hopping approach do invoke additional assumptions. The comparison of the mapping approach to these formulations may therefore (i) provide insight into the nature of these additional approximation and (ii) indicate whether the conceptual virtues of the mapping approach may be expected to result in practical advantages. 

All approaches for the description of nonadiabatic dynamics discussed so far have used the simple quasi-classical approximation (16) to describe the dynamics of the nuclear degrees of freedom. As a consequence, these methods are in general not able to account for processes or observables for which quantum effects of the nuclear degrees of freedom are important. Such processes include nuclear tunneling, interference effects in wave-packet dynamics, and the conservation of zero-point energy. In contrast to quasi-classical approximations, semiclassical methods take into account the phase exp iSi/h) of a classical trajectory and are therefore capable—at least in principle—of describing quantum effects. 

In contrast to the quasi-classical approaches discussed in the previous chapters of this review, Eq. (114) represents a description of nonadiabatic dynamics which is semiclassically exact in the sense that it requires only the basic semiclassical Van Vleck-Gutzwiller approximation [3] to the quantum propagator. Therefore, it allows the description of electronic and nuclear quantum effects. 

Even with the progress that has been made in rigorous quantum approaches, it is nevertheless possible to carry out such calculations only for relatively simple chemical systems. For example, the largest molecular system for which such calculations have been carried out is for the reaction H2 + OH — H2O + H. There is clearly interest, therefore, in the development of approximate versions of the approach that can be applied to more complex systems. Section III describes a semiclassical approximation for doing this, and Section IV concludes. 

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