OPEN THEORETICAL PROBLEMS

In spite of the great effort made in the last several decades, a large number of problems concerning the chemistry of phenols remain open wide for theoretical studies. 

The significance of the reaction of phenol with hydrogen has a number of important facets. First, the selective hydrogenation of phenol yields cyclohexanone, which is a key raw material in the production of both caprolactam for nylon 6 and adipic acid for nylon 6 . Second, due to the fact that phenol is an environmental toxin and phenolic waste has a variety of origins from industrial sources including oil refineries, petrochemical units, polymeric resin manufacturing and plastic units , catalytic hydrogenation of phenol is nowadays the best practicable environmental option .  

FIGURE 57. The complex of pentachlorophenol with 2-methylpyridine optimized at the B3LYP/6-31G(d) computational level. Bond lengths are in A, bond angles in deg 

The behaviour of the tyrosyl radicals involved in different processes and environments is not yet well understood Relatively little is known about the structure and selectivity of aryloxylium cations (Ar—0+) that are produced in the phenolic oxidation reactions and implicated in biological processes such as isoflavone synthesis . The thermochemistry which is relevant to the antioxidant properties of phenols as well as the solvent effects on their reactivity ° remain also a largely under-explored topic. Finally, the structure of phenol dimers and oligomers or even of some specific phenols also deserve more attention. We expect that these problems will be subjects for theoretical research in the coming years. 

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The investigation of defect patterns in oxide scales formed under a graded scale loading seems to be promising since critical load parameters could be analysed ex situ. However, there are a series of experimental difficulties and open theoretical problems in realising this approach. 

The intriguing formation of a few dioxopiperazines 12 opens new problems of competitive regiochemistry and stereochemistry. Related synthetic, x-ray, and theoretical studies are currently carried out (Ref. 18). 

This necessarily sketchy outline of the unsolved problems relating to this system is certainly not complete. We have mainly endeavored to give the reader a sampling of the variety of the open theoretical questions. We conclude this chapter with a small selection of the varied applications of this mathematically idealized theory for the description of thermod3mamic properties of real fluids. 

This section discusses some subtle difficulties that are glossed over in most of the treatments of electron transmission using the formalisms described above. These should be regarded as open theoretical issues that should be addressed in future developments. The source of these problems is our simplified treatment of what is actually a complex many-body open system. 

Theoretical Results. To solve the open-shell problem, various MO methods have been applied and, in some cases, compared with each other. Most commonly, the restricted Hartree-Fock formalism for the open-shell case (RHF) [15] and the unrestricted Hartree-Fock formalism before (UHF) [16] and after single spin annihilation (UHFASA) [17] have been used besides other methods, see [18 to 21] and footnote of Table 6, p. 234. 

To a large extent the opulent diversity of electronic and magnetic properties manifested by lanthanide materials arises from the existence of an open shell of atomic-like 4f electrons. A number of challenging theoretical problems accompanies the panoply of physical phenomena, not only for the exotic mixed valence systems but for comparitively simpler materials such as the metals as well. In this chapter we treat one of these issues, the calculation of 4f electron excitation energies in the lanthanide metals. Measurement of the excitation energies is one of the principal achievements of the high-energy spectroscopies which form the subject of the present volume. 

Semi-Theoretical and Empirical Velocity Fields Since the use of formulas to calculate the velocities outside an arbitrary opening could be very tedious, only some examples of these formulae are given. These calculations are best done on computers and there are some dedicated programs to calculate and visualize the flow fields outside exhaust openings. There could sometimes be problems when calculating the velocity field outside an opening close to... 

The relationships between thermodynamic entropy and Shannon s information-theoretic entropy and between physics and computation have been explored and hotly debated ever since. It is now well known, for example, that computers can, in principle, provide an arbitrary amount of reliable computation per kT of dissipated energy ([benu73], [fredkin82] see also the discussion in section 6.4). Whether a dissipationless computer can be built in practice, remains an open problem. We must also remember that computers are themselves physical (and therefore, ultimately, quantum) devices, so that any exploration of the limitations of computation will be inextricably linked with the fundamental limitations imposed by the laws of physics. 

Using the first-principles molecular-dynamics simulation, Munejiri, Shimojo and Hoshino studied the structure of liquid sulfur at 400 K, below the polymerization temperature [79]. They found that some of the Ss ring molecules homolytically open up on excitation of one electron from the HOMO to the LUMO. The chain-like diradicals S " thus generated partly recombine intramolecularly with formation of a branched Sy=S species rather than cyclo-Ss- Furthermore, the authors showed that photo-induced polymerization occurs in liquid sulfur when the Ss chains or Sy=S species are close to each other at their end. The mechanism of polymerization of sulfur remains a challenging problem for further theoretical work. 

For nonequilibrium statistical mechanics, the present development of a phase space probability distribution that properly accounts for exchange with a reservoir, thermal or otherwise, is a significant advance. In the linear limit the probability distribution yielded the Green-Kubo theory. From the computational point of view, the nonequilibrium phase space probability distribution provided the basis for the first nonequilibrium Monte Carlo algorithm, and this proved to be not just feasible but actually efficient. Monte Carlo procedures are inherently more mathematically flexible than molecular dynamics, and the development of such a nonequilibrium algorithm opens up many, previously intractable, systems for study. The transition probabilities that form part of the theory likewise include the influence of the reservoir, and they should provide a fecund basis for future theoretical research. The application of the theory to molecular-level problems answers one of the two questions posed in the first paragraph of this conclusion the nonequilibrium Second Law does indeed provide a quantitative basis for the detailed analysis of nonequilibrium problems. 

We have mentioned above the question of ring-opening in cyclopropane, as analyzed by EHT. In this case too the effect of the substituents is of paramount importance. A number of theoretical and experimental studies on this problem have been carried out (see (30) for a detailed account). EHT studies were done (30) for the compounds ... 

In metalloproteins, the paramagnet is an inseparable part of the native biomacromolecule, and so anisotropy in the metal EPR is not averaged away in aqueous solution at ambient temperatures. This opens the way to study metalloprotein EPR under conditions that would seem to approach those of the physiology of the cell more closely than when using frozen aqueous solutions. Still the number of papers describing metalloprotein bioEPR studies in the frozen state by far outnumbers studies in the liquid state. Several additional theoretical and practical problems are related to the latter (1) increased spin-lattice relaxation rate, (2) (bio)chemical reactivity, (3) unfavorable Boltzmann distributions, (4) limited tumbling rates, and (5) undefined g-strain. 

It is also worth emphasizing that recent theoretical work on photoinduced stepwise and concerted electron transfer/bond-breaking reactions opens the route to a more systematic combination than before of the electrochemical and photochemical approaches to the same problems. 

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