Classical descriptions

The classical description of magnetic resonance suffices for understanding the most important concepts of magnetic resonance imaging. The description is based upon the Bloch equation, which, in the absence of relaxation, may be written as... 

As in any field, it is usefiil to clarify tenninology. Tliroughout this section an atom more specifically refers to its nuclear centre. Also, for most of the section the /)= 1 convention is used. Finally, it should be noted that in the literature the label quantum molecular dynamics is also sometimes used for a purely classical description of atomic motion under the potential created by tlie electronic distribution. 

By using this approach, it is possible to calculate vibrational state-selected cross-sections from minimal END trajectories obtained with a classical description of the nuclei. We have studied vibrationally excited H2(v) molecules produced in collisions with 30-eV protons [42,43]. The relevant experiments were performed by Toennies et al. [46] with comparisons to theoretical studies using the trajectory surface hopping model [11,47] fTSHM). This system has also stimulated a quantum mechanical study [48] using diatomics-in-molecule (DIM) surfaces [49] and invoicing the infinite-onler sudden approximation (lOSA). 

The description of shock-compressed matter derived from physical and chemical observations, as presented in this book, is significantly different from that denved strictly from mechanical characteristics, which are the classical descriptions. This volume, with over 300 references and summaries of major review articles, provides a succinct introduction and critical analysis for scientists and engineers interested in the present state of shock-compression science. 

Classical descriptions of molecular phenomena can be remarkably successful, but we have to keep our eye on the intrinsic quantum nature of microscopic systems. 

A classical description of M can for example be a standard force field with (partial) atomic charges, while a quantum description involves calculation of the electronic wave function. The latter may be either a semi-empirical model, such as AMI or PM3, or any of the ab initio methods, i.e. HF, MCSCF, CISD, MP2 etc. Although the electrostatic potential can be derived directly from the electronic wave function, it is usually fitted to a set of atomic charges or multipoles, as discussed in Section 9.2, which then are used in the actual solvent model. 

The quasi-classical description of the Q-branch becomes valid as soon as its rotational structure is washed out. There is no doubt that at this point its contour is close to a static one, and, consequently, asymmetric to a large extent. It is also established [136] that after narrowing of the contour its shape in the liquid is Lorentzian even in the far wings where the intensity is four orders less than in the centre (see Fig. 3.3). In this case it is more convenient to compare observed contours with calculated ones by their characteristic parameters. These are the half width at half height Aa)i/2 and the shift of the spectrum maximum ftW—< > = 5a>+A, which is usually assumed to be a sum of the rotational shift 5larger scale A determined by vibrational dephasing. 

Here n is an operator of molecular axis orientation. In the classical description, it is just a unitary vector, directed along the rotator axis. Angle a sets the declination of the rotator from the liquid cage axis. Now a random variable, which is conserved for the fixed form of the cell and varies with its hopping transformation, is a joint set of vectors e, V, where V = VU...VL,.... Since the former is determined by a break of the symmetry and the latter by the distance between the molecule and its environment, they are assumed to vary independently. This means that in addition to (7.17), we have... 

A classical description of such a structure is of no real use. That is, if we attempt to describe the structure using the same tools we would use to describe a box or a sphere we miss the nature of this object. Since the structure is composed of a series of random steps we expect the features of the structure to be described by statistics and to follow random statistics. For example, the distribution of the end-to-end distance, R, follows a Gaussian distribution function if counted over a number of time intervals or over a number of different structures in space,... 

Consider reorientations of a diatomic surface group BC (see Fig. A2.1) connected to the substrate thermostat. By a reorientation is meant a transition of the atom C from one to another well of the azimuthal potential U(qi) (see Fig. 4.4)). The terminology used implies a classical (or at least quasi-classical) description of azimuthal motion allowing the localization of the atom C in a certain well. A classical particle, with the energy lower than the reorientation barrier Awhich does not interact with the thermostat cannot leave the potential well where it was located initially. The only pathway to reorientations is provided by energy fluctuations of a particle which arise from its contact with the thermostat. Let us estimate the average frequency of reorientations in the framework of this classical approach. 

In this Appendix, we attempted to elucidate the basic features of the transition of a particle from one potential well to another in various special cases at high reduced barriers p when a classical description is applicable, for p 1 when the probability for a rotational vibration to occur can be regarded as an estimate of the reorientation frequency for not too low temperatures, and, finally, for the low-temperature situation when subbarrier tunneling relaxation becomes dominant. However, the quantitative description of the processes considered cannot be taken as satisfactory, since it is rather fragmentary and, in addition, no general expression has been derived for an average reorientation frequency which would reduce to Eqs. (A2.4), (A2.26) or to (A2.38), (A2.39) in the above-listed special cases. As of now, an adequate approach has been developed which allows this quantity to be... 

If quantum mechanics is really the fundamental theory of our world, then an effectively classical description of macroscopic systems must emerge from it - the so-called quantum-classical transition (QCT). It turns out that this issue is inextricably connected with the question of the physical meaning of dynamical nonlinearity discussed in the Introduction. The central thesis is that real experimental systems are by definition not isolated, hence the QCT must be viewed in the relevant physical context. 

The (nonlocal) polarizabilities are important DFT reactivity descriptors. But, how are polarizabilities related to chemistry As stated above, an essential ingredient of the free energy surface is the potential energy surface and, in particular, its gradients. In a classical description of the nuclei, they determine the many possible atomic trajectories. Thanks to Feynman, one knows a very elegant and exact formulation of the force between the atoms namely [22,23]... 

In spite of the frequency-shifted excitation, the quantized PIP inevitably excites multiple sidebands located at n/At ( = 1, 2,...) from the centre band. An attempt was made16 to calculate the excitation profile of multiple bands created by a PIP of a constant RF field strength, using an approximate method based on the Fourier analysis. The accuracy of the method relies partly on the linear response of the spin system, which is, unfortunately, not true in most cases except for a small angle excitation. In addition, the spins inside a magnet consitute a quantum system, which is sensitive not only to the strengths but also to the phases of the RF fields. Any classical description is doomed to failure if the quantum nature of the spin system emerges. 

The classical description is quite different from the quantum. In classical dynamics we describe the coordinates and momenta simultaneously as a function of time and can follow the path of the system as it goes from reactants to products during the collision. These paths, called trajectories, provide a fnotion picture of collision process. The results of any real collision can be represented by computing a large number of trajectories to obtain distribution of post-collisions properties of interest (e.g. energy or angular distribution). In fact, the trajectory calculation means the transformation of one distribution function (reagent distribution, pre-collision) into another (product distribution, post-collision), which is determined by PE function. 

In cases where both the system under consideration and the observable to be calculated have an obvious classical analog (e.g., the translational-energy distribution after a scattering event), a classical description is a rather straightforward matter. It is less clear, however, how to incorporate discrete quantum-mechanical DoF that do not possess an obvious classical counterpart into a classical theory. For example, consider the well-known spin-boson problem—that is, an electronic two-state system (the spin) coupled to one or many vibrational DoF (the bosons) [5]. Exhibiting nonadiabatic transitions between discrete quantum states, the problem apparently defies a straightforward classical treatment. 

In this chapter, we are concerned with various theoretical formulations that allow us to treat nonadiabatic quantum dynamics in a classical description. To introduce the main concepts, we first give a brief overview of the existing methods and then discuss their application to ultrafast molecular photoprocesses. 

The goal of this chapter is twofold. First we wish to critically compare—from both a conceptional and a practical point of view—various classical and mixed quantum-classical strategies to describe non-Born-Oppenheimer dynamics. To this end. Section II introduces five multidimensional model problems, each representing a specific challenge for a classical description. Allowing for exact quantum-mechanical reference calculations, aU models have been used as benchmark problems to study approximate descriptions. In what follows, Section III describes in some detail the mean-field trajectory method and also discusses its connection to time-dependent self-consistent-field schemes. The surface-hopping method is considered in Section IV, which discusses various motivations of the ansatz as well as several variants of the implementation. Section V gives a brief account on the quantum-classical Liouville description and considers the possibility of an exact stochastic realization of its equation of motion. 

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