Structured molecules, classical mechanics

The question of predictability within the deterministic structure of classical mechanics was clearly appreciated by many eminent researchers in nonlinear systems theory and theoretical physics (see, e.g., Brillouin (I960)). Borel (1914) adds an additional twist to the predictability discussion. He argues that the displacement of a lump of matter with mass on the order of 1 g by as little as 1 cm and as far away as, e.g., the star Sirius is enough to preclude any prediction of the motion of the molecules of a volume of a classical gas for any longer than a firaction of a second, even if the initial conditions of the gas molecules are known with mathematical precision. Borel s example shows that many physical systems are not only sensitive to initial conditions, but also to miniscule changes in system parameters. The sensitivity to system parameters is a fundamental additional handicap for accurate long-time predictions. In the face of Borel s example, Brillouin (1960) points out that the prediction of the motion of gas molecules is not only very diflficult , as pointed... 

Molecular mechanics simulations use the laws of classical physics to predict the structures and properties of molecules. Molecular mechanics methods are available in many computer programs, including MM3, HyperChem, Quanta, Sybyl, and Alchemy. There are many different molecular mechanics methods. Each one is characterized by its particular/orce eW. A force field has these components ... 

Beyond the clusters, to microscopically model a reaction in solution, we need to include a very big number of solvent molecules in the system to represent the bulk. The problem stems from the fact that it is computationally impossible, with our current capabilities, to locate the transition state structure of the reaction on the complete quantum mechanical potential energy hypersurface, if all the degrees of freedom are explicitly included. Moreover, the effect of thermal statistical averaging should be incorporated. Then, classical mechanical computer simulation techniques (Monte Carlo or Molecular Dynamics) appear to be the most suitable procedures to attack the above problems. In short, and applied to the computer simulation of chemical reactions in solution, the Monte Carlo [18-21] technique is a numerical method in the frame of the classical Statistical Mechanics, which allows to generate a set of system configurations... 

The basic theories of physics - classical mechanics and electromagnetism, relativity theory, quantum mechanics, statistical mechanics, quantum electrodynamics - support the theoretical apparatus which is used in molecular sciences. Quantum mechanics plays a particular role in theoretical chemistry, providing the basis for the valence theories which allow to interpret the structure of molecules and for the spectroscopic models employed in the determination of structural information from spectral patterns. Indeed, Quantum Chemistry often appears synonymous with Theoretical Chemistry it will, therefore, constitute a major part of this book series. However, the scope of the series will also include other areas of theoretical chemistry, such as mathematical chemistry (which involves the use of algebra and topology in the analysis of molecular structures and reactions) molecular mechanics, molecular dynamics and chemical thermodynamics, which play an important role in rationalizing the geometric and electronic structures of molecular assemblies and polymers, clusters and crystals surface, interface, solvent and solid-state effects excited-state dynamics, reactive collisions, and chemical reactions. 

In the previous chapters, you have learned how to use DFT calculations to optimize the structures of molecules, bulk solids, and surfaces. In many ways these calculations are very satisfying since they can predict the properties of a wide variety of interesting materials. But everything you have seen so far also substantiates a common criticism that is directed toward DFT calculations namely that it is a zero temperature approach. What is meant by this is that the calculations tell us about the properties of a material in which the atoms are localized at equilibrium or minimum energy positions. In classical mechanics, this corresponds to a description of a material at 0 K. The implication of this criticism is that it may be interesting to know about how materials would appear at 0 K, but real life happens at finite temperatures. 

This is of relevance to the mechanism of carbonic anhydrase. This enzyme, which catalyzes the hydration of C02, has at its active site a Zn2+ ion ligated to the imidazole rings of three of its histidines. The classic mechanism for the reaction is that the fourth ligand is a water molecule which ionizes with a pKa of 7.37 The reactive species is considered to be the zinc-bound hydroxyl. Chemical studies show that zinc-bound hydroxyls are no exception to the rule of high reactivity. The H20 in structure 2.31 ionizes with a pKa of 8.7 and catalyzes the hydration of carbon dioxide and acetaldehyde.38... 

The basic principles behind classical mechanics are quite familiar to most of these students. Almost all of them have used F = ma, or can understand that a charge going around in a circle is a current. It is easy to use only these concepts to prove that something is wrong with any classical interpretation of atomic and molecular structure. Quantum mechanics allows us to predict the structure of atoms and molecules in a manner which agrees extremely well with experimental evidence, but the intrinsic logic cannot be understood without equations. 

Whereas the quantum-mechanical molecular Hamiltonian is indeed spherically symmetrical, a simplified virial theorem should apply at the molecular level. However, when applied under the Born-Oppenheimer approximation, which assumes a rigid non-spherical nuclear framework, the virial theorem has no validity at all. No amount of correction factors can overcome this problem. All efforts to analyze the stability of classically structured molecules in terms of cleverly modified virial schemes are a waste of time. This stipulation embraces the bulk of modern bonding theories. 

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