Classical behavior

Sometimes classical mechanics offers a sufficient description of the dynamical behavior of molecular sysfems. Such a description, however, does not provide the quantum energy levels that are involved in the postulate of equal probabilities. To apply in a classical mechanical framework, fhe postulate of statistical mechanics requires something analogous to quantum states and their energies. We consider the analogy to show that statistical mechanics can be applied without a quantum mechanical analysis however, the primary focus of this chapter uses quantum knowledge about molecules. 

The notion of an abstiact volume in the space of the coordinates of a Hamiltonian (i.e., position and momentum coordinates) is formalized by the introduction of phase space, of which there are at least two kinds. One kind should be regarded as a six-dimensional particle space. The six dimensions are those of the three spatial coordinates and the three momentum coordinates needed in the mechanical description of a single particle. At any instant in time, a particle is at one point in this six-dimensional phase space. If there were several particles in the system, each could be associated with a distinct point in this space at every instant in time. A six-dimensional box in phase space can be referred to as having a particular volume in that space. Then, the postulate of equal probabilities is a statement that the probability that the phase space point represents a single particle is the same in any one among all like-sized boxes. 

For a system of N particles, we can define a phase space of 6N coordinates, the position and momenta coordinates of each of the particles. In this second kind of phase space, the system as a whole is represented by a single point, whereas in the particle phase space, the system is represent by N points. 

The mechanical behavior of a system of N particles can be represented by a path or trajectory through the system s phase space. At some instant, the system is at one phase space point, and at a later instant, at another point. A line coimecting the points shows the evolution of the system in time. Under certain conditions, the behavior of the system can follow many different trajectories. Some volumes of phase space may be traversed by many of the possible trajectories, whereas other specific (position-momentum) volumes maybe traversed only a few times. We can therefore consider that there is a density of the tr ectory points in any particular volume of phase space. This density is a function of the 6N coordinates of the space. If we know the value of some dynamical variable at every point in phase space, we can obtain an average value for that variable by integrating the product of the dynamical variable and the density over the entire volume of the phase space. We can take as another postulate that such an average corresponds to the average behavior of the system. 

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Since 17 = 0 whenever P22 oi P21 are zero, the eorrcspoiidiiig classical behavior of eigenstate.s > and > must always involve a color change. In particular, the matrix form of the classical evolution is given by... 

It is worthwhile to first review several elementary concepts of reaction rates and transition state theory, since deviations from such classical behavior often signal tunneling in reactions. For a simple unimolecular reaction. A—>B, the rate of decrease of reactant concentration (equal to rate of product formation) can be described by the first-order rate equation (Eq. 10.1). 

First, consider the symmetric transition of a particle between unexcited vibrational states assuming classical behavior of the medium atoms which form the microstructure near the tunneling particle and determine its potential energy. The states of the system corresponding to the localization of the particle in the initial and... 

Dendrimers possessing terminal phosphino groups but also other functional chain ends such as allylamine present the same classical behavior towards Fe2(CO)9 and W(CO)5(THF) [21]. 

Complications that arise with this simple reaction are twofold. First, because of the low mass of the hydrogen atom its movement frequently exhibits non-classical behavior, in particular quantum-mechanical tunneling, which contributes significantly to the observed kinetic isotope effect, and in fact dominates at low temperature (Section 6.3). Secondly, in reaction 10.2 protium rather than deuterium transfer may occur ... 

At high F, when the spacing of vibrational energy levels is low with respect to thermal energy, crystalline solids begin to show the classical behavior predicted by kinetic theory, and the heat capacity of the substance at constant volume (Cy) approaches the theoretical limit imposed by free motion of all atoms along three directions, in a compound with n moles of atoms per formula unit limit of Dulong and Petit) ... 

It is found that for data at each temperature, data collapse occurs for time scaling by as shown in Figure 2. This is consistent both with the classical model of Mullins, as well as the pinch-off model. The difference between these two models is the temperature dependence of the relaxation rate. For classical behavior, one expects the characteristic time to scale according to... 

One large approximation is the use of Eq. (8.4) for the interelectronic repulsion, since it ignores the energetic effects associated with correlation and exchange. It is useful to introduce the concept of a hole function , which is defined so that it corrects for the energetic errors introduced by assuming classical behavior. In particular, we write... 

Classical behavior is expected to occur on time scales that are shorter than the Heisenberg time (3.4). As the energy of the initial wavepacket increases, the quasiclassical and semiclassical regimes will extend over longer time intervals if the level density increases, as is usually the case in bounded... 

Although mi completely satisfactory single theory of liquid helium has yet been formulated, one can say that most of the remarkable properties are qualitatively understood and are due 10 Ihe predominance nl quantum effects, including the dillerence in the statistics of the even and odd isotopes. Titus helium is the one example in nature of a quantum liquid, ail olher liquids showing only minor deviations from classical behavior. 

F(k, t) and Fs(k, t), it should be noted, are one-sided quantum-mechanical time-correlation functions. We shall be interested in the classical behavior... 

The bare Coulombic interaction (p = 1) and interactions of charges with rotating dipoles (p = 4) do not fall into this class, and it has been argued for a long time [30] that in this case one expects analytical ( classical ) behavior. This implies that the system can be described by a mean-field Hamiltonian, in which the interaction of a particle is ascribed to the mean field of all other particles, thus ignoring local fluctuations [10]. In real ionic fluids the... 

Starting with a study on the liquid-vapor coexistence of ammonium chloride (NH4C1) [34], there have been repeated reports on classical ionic criticality [4], but none of these studies allows unambiguous conclusions [14]. In 1990 more decisive results were reported by Singh and Pitzer [35], who observed a parabolic liquid-liquid coexistence curve of an electrolyte solution. This apparent classical behavior was the stimulus for most theoretical and experimental work reported here. 

Ac susceptibility measurements performed on the sample with the smallest clusters in a bias field are shown in Fig. 14. x has a maximum that shifts to higher temperature as the bias field increases, while that of x decreases. This apparently contradicts the classical behavior since the bias field reduces the activation energy. This bizarre feature is clarified by the measurements on the larger particles at sufficiently high field (300 Oe), since in that case the maximum, shifted to higher temperature, is not frequency dependent (Fig. 15). Under these conditions, the x response is dominated by the equilibrium susceptibility, and the temperature at the maximum is the temperature of the equilibrium susceptibility Teq 2/uII/kn,... 

The adiabatic approximation means the neglect of the nuclear motion in the Schrodinger equation. The electronic structure is thus calculated for a set of fixed nuclear coordinates. This approach can in principle be exact if one uses the set of wave functions for fixed nuclear coordinates as a basis set for the full Schrodinger equation, and solves the nuclear motion on this basis. The adiabatic approximation stops at the step before. (The Born-Oppenheimer approximation assumes a specific classical behavior of the nuclei and hence it is more approximate than the adiabatic approximation.)... 

This microscopic interaction model can be used to explain more specific interactions between drug molecules and lipids. Such specific interactions could be a selective coupling between a drag molecule and a particular chain conformation of the lipid (kink excitation). This could have a dramatic effect on the fluctuation system. The drug molecule would then control the formation of interfaces between lipid domains and bulk phase in the neighborhood of the transition. First results on an extended model of this type [50] have confirmed this view and demonstrated that the partition coefficient can develop non-classical behavior by displaying a maximum near the transition. And such a maximum has in fact been observed experimentally... 

The two-electron integrals (Equation 6.32) are determined from atomic experimental data in the one-center case, and are evaluated from a semiempirical multipole model in the two-center case that ensures correct classical behavior at large distances and convergence to the correct one-center limit. Interestingly, this parameterization results in damped effective electron-electron interactions at small and intermediate distances, which reflects a (however less regular) implicit partial inclusion of electron correlation (Thiel, 1998). In this respect, semiempirical methods go beyond the HF level, and may accordingly be superior to HF ab initio treatments for certain properties that have a direct or indirect connection to the parameterization procedure. 

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