Minimum motion principle

As can be seen, this index attains its maximal value of unity for two identical structures (9 = 9,

monotonously decreases. The use of this index for the formulation of the least motion principle arises from the following simple idea. Let us assume that we are on a reaction path at point characterised by the wave function P(, q> ) and we are looking for such an infinitesimally close structure (9, (p) for which the transformation (9,

requires minimal change in electronic configuration. This condition is equivalent to a search of the direction along which the derivative of K at the point = 9 and q> = q> attains its minimum. This directional derivative can be mathematically described as (27),... 

In this chapter, we review our latest results on the validity of the maximum hardness and minimum polarizability principles in nontotally symmetric vibrations. These nuclear displacements are particularly interesting because they keep the chemical and external potentials approximately constant, thus closely following the two conditions of Parr and Chattaraj required for the strict compliance with the maximum hardness principle. We show that, although these principles are obeyed by most nontotally symmetric vibrations, there are some nontotally symmetric displacements that refuse to comply with them. The underlying physical reasons for the failure of these two principles in these particular nuclear motions are analyzed. Finally, the application of this breakdown to detect the most aromatic center in polycyclic aromatic hydrocarbons is discussed. 

Enzyme catalysts typically have low activation energies. The synchronized action of multi-atom displacements to optimize the interaction energy with the pretransition-state complex imphes a decrease in the entropic state of the complex. This has been analyzed by Liktensteinl l, who formulated the principle of optimum motion , in contrast to the principle of minimum motion . 

From now on, we assume that the geometry under study eorresponds to that of a stable minimum about whieh vibrational motion oeeurs. The treatment of unstable geometries is of great importanee to ehemistry, but this Chapter deals with vibrations of stable speeies. For a good treatment of situations under whieh geometrieal instability is expeeted to oeeur, see Chapter 2 of the text Energetic Principles of Chemical Reactions by... 

The most celebrated textual embodiment of the science of energy was Thomson and Tait s Treatise on Natural Philosophy (1867). Originally intending to treat all branches of natural philosophy, Thomson and Tait in fact produced only the first volume of the Treatise. Taking statics to be derivative from dynamics, they reinterpreted Newton s third law (action-reaction) as conservation of energy, with action viewed as rate of working. Fundamental to the new energy physics was the move to make extremum (maximum or minimum) conditions, rather than point forces, the theoretical foundation of dynamics. The tendency of an entire system to move from one place to another in the most economical way would determine the forces and motions of the various parts of the system. Variational principles (especially least action) thus played a central role in the new dynamics. 

Evans and Baranyai [51, 52] have explored what they describe as a nonlinear generalization of Prigogine s principle of minimum entropy production. In their theory the rate of (first) entropy production is equated to the rate of phase space compression. Since phase space is incompressible under Hamilton s equations of motion, which all real systems obey, the compression of phase space that occurs in nonequilibrium molecular dynamics (NEMD) simulations is purely an artifact of the non-Hamiltonian equations of motion that arise in implementing the Evans-Hoover thermostat [53, 54]. (See Section VIIIC for a critical discussion of the NEMD method.) While the NEMD method is a valid simulation approach in the linear regime, the phase space compression induced by the thermostat awaits physical interpretation even if it does turn out to be related to the rate of first entropy production, then the hurdle posed by Question (3) remains to be surmounted. 

Following Atkins [68], the propagation of particles follows a path dictated by Newton s laws, equivalent to Hamilton s principle, that particles select paths between two points such that the action associated with the path is a minimum. Therefore, Fermat s principle for light propagation is Hamilton s principle for particles. The formal definition of action is an integral identical in structure with the phase length in physical optics. Therefore, particles are associated with wave motion, the wave-particle dualism. Hamilton s principle of least... 

The primary results of these first experiments from each facility were primarily proof-of-principle. Hussey et al.9 used a MCP detector with intrinsic spatial resolution of 25 pm and overall image resolution of about 30 pm. With this setup, a test section was operated in a differential cell mode, with a minimum stoichiometric ratio of about 50 on both the anode and cathode. Due to cell motion, it was not possible to quantify the total water content in the cell, but relative changes in the through-plane water content were observed from open circuit voltage, and the water content increased as a function of current density. 

In terms of quantum mechanics, a system with zero energy is impossible. A quantum system must possess a minimum energy of Ev. This postulate is due to the irrepressible zero-point motion imposed on microscopic systems by the uncertainty principle and by quantization. Thus, the classical concept of nuclei in space and associated motion is replaced by the concept of a nuclear or vibrational... 

The possibility that reacting species prefer to react along those paths in which they undergo the least modification has always been intuitively attractive. At one time or another, so-called principles of minimum structural change or deformation, configurational change, and minimum atomic and electronic motion have been invoked (Wheland, 1960 Hine, 1966). To account for Michael s rule of favored anti 1,2-addition, Pfeiffer formulated acetylenes as tram-heat structures in 1904 Frankland (1912) suggested that anti elimination is favored by an inherent tendency to centric symmetry. The more conscious applications of PLM by Muller after 1886, are probably misapplications of the principle, since they were usually concerned with complex pyrolytic reactions above 1000° (Muller and Peytral, 1924). 

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