Classical phase

A situation that arises from the intramolecular dynamics of A and completely distinct from apparent non-RRKM behaviour is intrinsic non-RRKM behaviour [9], By this, it is meant that A has a non-random P(t) even if the internal vibrational states of A are prepared randomly. This situation arises when transitions between individual molecular vibrational/rotational states are slower than transitions leading to products. As a result, the vibrational states do not have equal dissociation probabilities. In tenns of classical phase space dynamics, slow transitions between the states occur when the reactant phase space is metrically decomposable [13,14] on the timescale of the imimolecular reaction and there is at least one bottleneck [9] in the molecular phase space other than the one defining the transition state. An intrinsic non-RRKM molecule decays non-exponentially with a time-dependent unimolecular rate constant or exponentially with a rate constant different from that of RRKM theory. 

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 quantuin-seiniclassical 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 seiniclassical 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]. 

Figure 15. Left-. Geometry of the surface 6 = 0 in Eq. (46) with fixed total angular momenta S and N. Properties of the special points A, B, C, and D are listed in Table I. All other permissible classical phase points lie on or inside the surface of the rounded tetrahedron. Right. Critical section at J. Continuous lines are energy contours for y = 0.5 and N/S = 4. Dashed lines are tangents to the section at D. Axes correspond to normalized coordinates, /NS and K JiN + S). Taken from Ref. [2] with permission of Elsevier.

Fig. 2.2 Landmark events in the cell cycle of Saccharomyces cerevisiae. Gl, S, G2 and M are the classical phases of the eukaryotic cell cycle.

In quantum statistical mechanics where a density operator replaces the classical phase density the statistics of the grand canonical ensemble becomes feasible. The problem with the classical formulation is not entirely unexpected in view of the fact that even the classical canonical ensemble that predicts equipartitioning of molecular energies, is not supported by observation. 

Integration of the phase density over classical phase space corresponds to finding the trace of the density matrix in quantum mechanics. Transition to a new basis is achieved by unitary transformation... 

Limitation to ensembles that allow exchange of energy, but not of matter, with their environment is unnecessarily restrictive and unrealistic. What is required is an ensemble for which the particle numbers, Nj also appear as random variables. As pointed out before, the probability that a system has variable particle numbers N and occurs in a mechanical state (p, q) can not be interpreted as a classical phase density. In quantum statistics the situation is different. Because of second quantization the grand canonical ensemble, like the microcanonical and canonical ensembles, can be represented by means of a density operator in Hilbert space. 

Figure 3. Classical phase portraits (upper panel), residual quantum wavefunctions (middle panel), and ionization probability versus time (in units of the period T) (bottom panel). The parameters are (A) F = 5.0, iv = 0.52 (B) F = 20, iv = 1.04 and (C) F = 10 and u> = 2.0. Note that the peak structure of the final wavefunction reflects both stable and unstable classical fixed points. For case C, the peaks are beginning to coalesce reflecting the approach of the single-well effective potentiai (see text). 

The results of the quantum simulations for cases A, B and C are shown in the lower two panels in Fig. 3. The corresponding classical phase portraits shown reinforce our inferences from the stability diagram no stabilization for A while larger islands exist for C as compared with B. However, the ionized fraction as calculated from the quantum evolution supports the contrary result that there is more stabilization for A as compared with B. Case C is the most stable which is at least consistent with the classical prediction. What is the origin of this discrepancy ... 

The distribution generated by surprisal analysis is meant to reproduce the results of actual interest, a typical example being the distribution of vibrational energy of products, which is of interest say for chemical laser action. [3] The distribution is not meant to reproduce the fully detailed distribution in classical phase space, which, as already noted, has a to be a highly correlated and complicated distribution. 

Second, the mapping approach to nonadiabatic quantum dynamics is reviewed in Sections VI-VII. Based on an exact quantum-mechanical formulation, this approach allows us in several aspects to go beyond the scope of standard mixed quantum-classical methods. In particular, we study the classical phase space of a nonadiabatic system (including the discussion of vibronic periodic orbits) and the semiclassical description of nonadiabatic quantum mechanics via initial-value representations of the semiclassical propagator. The semiclassical spin-coherent state method and its close relation to the mapping approach is discussed in Section IX. Section X summarizes our results and concludes with some general remarks. 

That is, the classical DoF propagate according to a mean-field potential, the value of which is weighted by the instantaneous populations of the different quantum states. A MFT calculation thus consists of the self-consistent solution of the time-dependent Schrodinger equation (28) for the quantum DoF and Newton s equation (32) for the classical DoF. To represent the initial state (15) of the molecular system, the electronic DoF dk Q) as well as the nuclear DoF xj Q) and Pj 0) are sampled from a quasi-classical phase-space distribution [23, 24, 26]. 

Recently, Stock and Muller have proposed an alternative strategy to tackle the ZPE problem [102, 103, 224]. The theory is based on the observation that the unphysical flow of ZPE is a consequence of the fact that the classical phase-space distribution may enter regions of phase space that correspond to a... 

In order to obtain thin skinned, high flux membranes from PVA, several approaches were tried. The method presented in this article is reminiscent of the classical phase inversion method, which is widely applied in casting of asymmetric RO membranes. However, instead of using a gelling bath composed of a nonsolvent... 

The mixture CMC is plotted as a function of monomer composition in Figure 1 for an ideal system. Equation 1 can be seen to provide an excellent description of the mixture CMC (equal to Cm for this case). Ideal solution theory as described here has been widely used for ideal surfactant systems (4.6—18). Equation 2 can be used to predict the micellar surfactant composition at any monomer surfactant composition, as illustrated in Figure 2. This relation has been experimentally confirmed (ISIS) As seen in Figure 2, for an ideal system, if the ratio XA/yA < 1 at any composition, it will be so over the entire composition range. In classical phase equilibrium thermodynamic terms, the distribution coefficient between the micellar and monomer phases is independent of composition. 

A wide range of melt compositions undergo phase separation upon cooling. Two mechanisms are observed classical phase separation (dashed curve) as well as by spinodal (dot-dashed curve) mechanisms. Both the position of these regions and the onset temperature of phase separation reported in the literature vary (e.g., Roth et al. 1987), but there is no doubt that the phenomenon occurs amongst the glassy phase of aluminosilicate fly ash despite the presence of other oxides that tend to promote miscibility. 

The term Counter Phase Transfer Catalysis (CPTC) was coined by Okano214,215 to describe biphasic reactions catalysed by water soluble transition metal complexes which involve transport of an organic-soluble reactant into the aqueous phase where the catalytic reaction takes place. Similarly, Mathias and Vaidya564,565 gave the name Inverse Phase Transfer Catalysis to describe this kind of biphasic catalysis which contrasts with classical Phase Transfer Catalysis where the reaction occurs in the organic phase and does not involve formation of micelles.389,564... 

Classical Phase Transfer assisted Organometallic Catalysis588"601 is a further important field which has found industrial applications e.g. in the carbonylation of benzyl chloride to phenylacetic acid using NaCo(CO)4/Bu4NBr catalysts in aqueous NaOH practiced by Montedison.52,466,589,596,601 However, a detailed discussion of classical phase transfer catalysis is beyond the scope of this chapter which is devoted to systems in which the catalytic conversion takes place in the aqueous phase. 

While classical phase diagrams provide a powerful methodology for grasping the thermodynamic behavior of few-component systems, it is evident that the restricted 2D or 3D realm of human graphical intuition cannot adequately cope with the complexities of many-component systems. Hence, it is important to find generalized analytical techniques that can accurately represent many-component phase behavior for arbitrary values of c. Such techniques will be considered in the metric geometric representation of multicomponent phenomena (Chapter 12). 

By contrast, in heavy-light-heavy molecules such as HMuH, C1HC1, or IHI, a very extended elliptic island exists in the classical phase space [150]. In such cases, the elliptic island may be the support of several metastable states that can be obtained by Bohr-Sommerfeld quantization. Their lifetime is determined by dynamical tunneling from inside the elliptic island to the outside regions. 

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