Semi-classical approaches

Finally, semi-classical approaches to non-adiabatic dynamics have also been fomuilated and siiccessfLilly applied [167. 181]. In an especially transparent version of these approaches [167], one employs a mathematical trick which converts the non-adiabatic surfaces to a set of coupled oscillators the number of oscillators is the same as the number of electronic states. This mediod is also quite accurate, except drat the number of required trajectories grows with time, as in any semi-classical approach. 

Nuclear spin relaxation is considered here using a semi-classical approach, i.e., the relaxing spin system is treated quantum mechanically, while the thermal bath or lattice is treated classically. Relaxation is a process by which a spin system is restored to its equilibrium state, and the return to equilibrium can be monitored by its relaxation rates, which determine how the NMR signals detected from the spin system evolve as a function of time. The Redfield relaxation theory36 based on a density matrix formalism can provide... 

We can conclude that the present method of correcting TF calculations provides adequate estimations of expectation values for ground state atoms taking into account the simplicity of the model and it self-consistent nature, where no empirical parameters are used. It provides information about the asymptotic behaviour of quantities such as p(0) and (r 2) that cannot be evaluated with the standard semi classical approach and allow us to estimate momentum expectation values which are not directly related to the density in an exact way. 

We have therefore shown that adiabatic surfaces can be said to cross off the real coordinate axes, and indeed if the classical equations of motion are solved in complex coordinate space then it is possible to simulate non-adiabatic processes. This can be considered as the basis of the Stuckelberg semi-classical approach to non-adiabatic transitions in atom-atom collisions (64) and it has been recently extended to more degrees of freedom (65). Moreover the actual form of potential surfaces in the complex plane has been obtained by direct calculation (66). 

In 1974, Polinger [19] used the WKB method for the E e JT problem. Recently, it was also used in the explanation of the non-degenerate ground state for the same problem [4]. Further discussions were given by O Brien [11] on the semi-classical method for both one-dimensional and multi-dimensional JT problems. All these semi-classical approaches simplified the multi-dimensional cases to onedimensional problems. 

The approach which will be reviewed here has been formulated within the framework of the quantum mechanical polarizable continuum model (PCM) [7], Within this method, the effective properties are introduced to connect the outcome of the quantum mechanical calculations on the solvated molecules to the outcome of the corresponding NLO experiment [8], The correspondence between the QM-PCM approach and the semi-classical approach will also be discussed in order to show similarities and differences between the two approaches. 

O. Tapia, Beyond standard quantum chemical semi-classic approaches Towards a quantum theory of enzyme catalysis, in P. Paneth, A. Dybala-Defratyka (Eds.), Kinetics and Dynamics, From nano- to bio-scale, Challenges and advances in computational chemistry and physics 12, Springer Science, Dordrecht, 2010, p. 267-298. 

In the semi-classical approach [18], the golden-rule type expression is used (Eq. 2) in which the rate is a product of an electronic matrix element squared... 

Fig. 1. The classical (a) and the semi-classical (b-d) representations of the Marcus theory for X — 1.0 eV at T = 300 K. In the classical expression (Eq. 1), X determines both the position of the maximum and the breadth of the parabola. The maximum keI is determined by the frequency factor (Z, here taken as 6 x 10 1 s ) in the Eyring expression (ket = KZexp( — AGlJkbT) where k is the transmission coefficient, usually taken to be unity). In the semi-classical approach the reorganization energy is explicitly divided into Xh (here equal 0.2 eV) and 2a (0.8 eV). The value of V is chosen to...

In the semi-classical approach each point of the potential energy surface is regarded as a state, weighed by a Boltzmann distribution. The probability... 

In the semi-classic approach. Group Theory for Non-rigid molecules allows to select the isoenergetic conformations, and to save calculation time. 

Marangoni s group subsequently used a thermodynamic approach to modify the model (Marangoni 2000 Marangoni and Rogers 2003). For spherical clusters using a van der Waals forces approach or a semi-classical approach based on the bluk properties of oils, the Young modulus ( ) becomes, respectively ... 

As mentioned above, the basic theory of the Raman effect was developed before its discovery. However, at this time numerical calculations of the intensity of Raman lines were impossible, because these require information on all eigenstates of a scattering system. Placzek (1934) introduced a semi-classical approach in the form of his polarizability theory. This provided a basis for many other theoretical and experimental studies. Physicists and chemists worldwide realized the importance of the Raman effect as a tool for qualitative and quantitative analysis and for the detennination of structure. 

In the keV energy domain, where experiments are available on both examples, semi-classical approaches using the EIKONXS code, based on an efficient propagation method [29], may be used with a good accuracy. 

It is the purpose of this paper to examine the energy conservation requirement for nonlinear light-matter interactions, in general, and for passive processes that contain resonances, in particular. The semi-classical approach, in which the radiation fields satisfy the classical Maxwell s equations, is used to define Wp in terms of the nonlinear electrical... 

Semi-classical Approaches the SCDS-Pixel Method... 

Chapter 6 introduces the product operator formalism for analysing NMR experiments. This approach is quantum mechanical, in contrast to the semi-classical approach taken by the vector model. We will see that the formalism is well adapted to describing pulsed NMR experiments, and that despite its quantum mechanical rigour it retains a relatively intuitive approach. Using product operators we can describe important phenomena such as the evolution of couplings during spin echoes, coherence transfer and the generation of multiple quantum coherences. 

In their simulations, L6vesque et used a standard Leimard-Jones interaction potential between hydrogen molecules, and included the effect of quadrupolar interactions by adding a Coulomb interaction term in which each hydrogen molecule is represented by three effective charges q (q = 0.4829e at the position of the protons and q = -e at the centre of mass of the molecule). The adsorbate-adsorbent interaction was modeled with a standard Lennard-Jones potential. In order to partially account for quantum effects at 77 K, a semi-classical approach based on the Feynman-Hibbs effective potential was used ... 

Quantum-mechanical expressions for the polarizability and other higher-order molecular response tensors are obtained by taking expectation values of the operator equivalent of the electric dipole moment (2.5) using molecular wavefunctions perturbed by the light wave (2.4). This particular semi-classical approach avoids the complications of formal time-dependent perturbation theory it has a respectable pedigree, being found in Placzek s famous treatise on the Raman effect [9], and also in the books by Born and Huang [lO] and Davydov [ll]. Further details of the particular version outlined here can be found in my own book [12]. 

Essentially, this proposal relaxes the original idea about the preferred pathways, yet in an elaborate fashion. Unfortunately, this is still a qualitative (semi-)classical model requiring much research work yet to be done it is therefore hard to predict the success of this approach. For example, one may notice that this proposal does not substantially go beyond the standard pathways (semi-)classical approach to the issue, especially in its kinematical context. ... 

Why should one believe in quantum-mechanical behavior of the large molecules After all, the (semi-) classical approach seems perfectly to work for most purposes in chemistry. 

Needless to say, the system S (the conformation ) is (likewise in the (semi-)classical approach) a characteristic of a (single) molecule as a whole. That is, as usual, we do not take into account the local details of the conformational rotations themselves, which essentially take into account the electron-state transitions. As much as we can see, these are of the secondary importance to our model, which abandons the concept of the transitions in T-space. Abandoning the T-space is key to the possible success of our model. It is a decoherence-Iike process that breaks stability of conformations (in the nonstationary state), eventually giving rise to the possibility of rather fast conformational transitions. 

The LevinthaTs paradox is an open problem still. To avoid the core of the problem — it s kinema-tical aspect — we propose a new approach in this regard. Actually, we treat the macromolecules conformations as the quantum-mechanical observable. Bearing in mind the foundations of the decoherence theory, we are able to model both, existence and maintenance of the conformations as well as the conformational transitions in the rather short time intervals. Our model is rather qualitative yet a general one — while completely removing the LevinthaTs paradox — in contradistinction with the (semi-)classical approach to the issue. 

Senyshyn et al. (2005b) calculated the above-mentioned properties and determined the thermal expansion coefficient of rare earth gallates using a semi-classical approach. Ideal (X-ray) density, Griineisen parameter, isohoric heat capacity Cy, bulk and shear moduli, and thermal expansion coefficient were calculated for RGaOa (R = La-Gd) at 300 K are listed in Table 47. 

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