Some Useful Integral Representations for Information-Theoretic Analyses

This work is an extension of our earlier article, where a well-known integral representation of the logarithmic function was explored and was accompanied with demonstrations of its usefulness in obtaining compact, easily-calculable, exact formulas for quantities that involve expectations of the logarithm of a positive random variable. Here, in the same spirit, we derive an exact integral representation (in one or two dimensions) of the moment of a nonnegative random variable, or the sum of such independent random variables, where the moment order is a general positive non-integer real (also known as fractional moments). The proposed formula is applied to a variety of examples with an information-theoretic motivation, and it is shown how it facilitates their numerical evaluations. In particular, when applied to the calculation of a moment of the sum of a large number, n, of nonnegative random variables, it is clear that integration over one or two dimensions, as suggested by our proposed integral representation, is significantly easier than the alternative of integrating over n dimensions, as needed in the direct calculation of the desired moment.     

Superlattice Turing structures in a photosensitive reaction-diffusion system

Families of complex superlattice structures, consisting of combinations of basic hexagonal or square patterns, are found in a photosensitive reaction-diffusion system. The structures are induced by simple illumination patterns whose wavelengths are appropriately related to that of the system's intrinsic Turing pattern. Computer simulations agree with the structures and their stability. The technique offers a general approach to generating superlattices for use in information storage and other applications.     

Identity SN2 reactions X- + CH3X --> XCH3 + X- (X=F, Cl, Br, and I) in vacuum and in aqueous solution: a valence bond study

The recently developed (L. Song, W. Wu, Q. Zhang, S. Shaik, J. Phys. Chem. A 2004, 108, 6017) valence bond method coupled with a polarized continuum model (VBPCM) has been applied to the identity SN2 reaction of halides in the gas phase and in aqueous solution. The barriers computed at the level of the breathing orbital VB method (P. C. Hiberty, J. P. Flament, E. Noizet, Chem. Phys. Lett. 1992, 189, 259), BOVB and VBPCM//BOVB, are comparable to CCSD(T) and CCSD(T)//PCM results and to experimentally derived barriers in solution (W. J. Albery, M. M. Kreevoy, Adv. Phys. Org. Chem. 1978, 16, 85). The reactivity parameters needed to apply the valence bond state correlation diagram (VBSCD) method (S. Shaik, J. Am. Chem. Soc. 1984, 106, 1227), were also determined by VB calculations. It has been shown that the reactivity parameters along with their semiempirical derivations provide a satisfactory qualitative and quantitative account of the barriers.     

Correlating DFT-Calculated Energy Barriers to Experiments in Nonheme Octahedral Fe(IV) O Species

The experimentally measured bimolecular reaction rate constant, k(2) , should in principle correlate with the theoretically calculated rate-limiting free energy barrier, ΔG(≠) , through the Eyring equation, but it fails quite often to do so due to the inability of current computational methods to account in a precise manner for all the factors contributing to ΔG(≠) . This is further aggravated by the exponential sensitivity of the Eyring equation to these factors. We have taken herein a pragmatic approach for C?H activation reactions of 1,4-cyclohexadiene with a variety of octahedral nonheme Fe(IV) O complexes. The approach consists of empirically determining two constants that would aid in predicting experimental k(2) values uniformly from theoretically calculated electronic energy (ΔE(≠) ) values. Shown in this study is the predictive power as well as insights into energy relationships in Fe(IV) O C?H activation reactions. We also find that the difference between ΔG(≠) and ΔE(≠) converges at slow reactions, in a manner suggestive of changes in the importance of the triplet spin state weight in the overall reaction.     

Orbitals and the Interpretation of Photoelectron Spectroscopy and (e,2e) Ionization Experiments

Electron momentum spectroscopy, scanning tunneling microscopy, and photoelectron spectroscopy provide unique information about electronic structure, but their interpretation has been controversial. This essay discusses a framework for interpretation. Although this interpretation is not new, we believe it is important to present this framework in light of recent publications. The key point is that these experiments provide information about how the electron distribution changes upon ionization, not how electrons behave in the pre‐ionized state. Therefore, these experiments do not lead to a “selection of the correct orbitals” in chemistry and do not overturn the well‐known conclusion that both delocalized molecular orbitals and localized molecular orbitals are useful for interpreting chemical structure and dynamics. The two types of orbitals can produce identical total molecular electron densities and therefore molecular properties. Different types of orbitals are useful for different purposes.     

Compound I in heme thiolate enzymes: a comparative QM/MM study

This study directly compares the active species of heme enzymes, so-called Compound I (Cpd I), across the heme-thiolate enzyme family. Thus, sixty-four different Cpd I structures are calculated by hybrid quantum mechanical/molecular mechanical (QM/MM) methods using four different cysteine-ligated heme enzymes (P450(cam), the mutant P450(cam)-L358P, CPO and NOS) with varying QM region sizes in two multiplicities each. The overall result is that these Cpd I species are similar to each other with regard to many characteristic features. Hence, using the more stable CPO Cpd I as a model for P450 Cpd I in experiments should be a reasonable approach. However, systematic differences were also observed, and it is shown that NOS stands out in most comparisons. By analyzing the electrical field generated by the enzyme on the QM region, one can see that (a) the protein exerts a large influence and modifies all the Cpd I species compared with the gas-phase situation and (b) in NOS this field is approximately planar to the heme plane, whereas it is approximately perpendicular in the other enzymes, explaining the deviating results on NOS. The calculations on the P450(cam) mutant L358P show that the effects of removing the hydrogen bond between the heme sulfur and L358 are small at the Cpd I stage. Finally, Mossbauer parameters are calculated for the different Cpd I species, enabling future comparisons with experiments. These results are discussed in the broader context of recent findings of Cpd I species that exhibit large variations in the electronic structure due to the presence of the substrate.     

Can a single oxidant with two spin states masquerade as two different oxidants? A study of the sulfoxidation mechanism by cytochrome p450

Density functional calculations were performed on the sulfoxidation reaction by a model compound I (Cpd I) of cytochrome P450. By contrast to previous alkane hydroxylation studies, which exhibit a dominant low-spin (LS) pathway, the sulfoxidation follows a dominant high-spin (HS) reaction. Thus, competing hydroxylation and sulfoxidation processes as observed for instance by Jones et al. (Volz, T. J.; Rock, D. A.; Jones, J. P. J. Am. Chem. Soc. 2002, 124, 9724) are the result of a two-state reactivity scenario, whereby the hydroxylation originates from the LS pathway and the sulfoxidation from the HS pathway. In this manner, two spin states of a single oxidant (Cpd I) can be disguised as two different oxidants. The calculations rule out the possibility that a second oxidant (the ferric peroxide, Cpd 0 species) interferes in the observed results of Jones et al.     

Active species of horseradish peroxidase (HRP) and cytochrome P450: two electronic chameleons

The active site of HRP Compound I (Cpd I) is modeled using hybrid density functional theory (UB3LYP). The effects of neighboring amino acids and of environmental polarity are included. The low-lying states have porphyrin radical cationic species (Por(*)(+)). However, since the Por(*)(+) species is a very good electron acceptor, other species, which can be either the ligand or side chain amino acid residues, may participate in electron donation to the Por(*)(+) moiety, thereby making Cpd I behave like a chemical chameleon. Thus, this behavior that was noted before for Cpd I of P450 is apparently much more wide ranging than initially appreciated. Since chemical chameleonic behavior property was found to be expressed not only in the properties of Cpd I itself, but also in its reactivity, the roots of this phenomenon are generalized. A comparative discussion of Cpd I species follows for the enzymes HRP, CcP, APX, CAT (catalase), and P450.