Electron transfer in macromolecules: Green's function and diagrammatic techniques (continued fraction representation)

In order to study the interfacial electron transfer, selfassembled monolayers (SAMs) on electrode surface have been attractive as a model system because they provide a stable and structurally well-defined monolayer with an adjustable thickness and desirable function. This characteristic of SAMs affords an opportunity to study fundamental issues such as the effects of distance and interfacial structure on the long-range electron transfer kinetics between a redox active species and an electrode. Weaver and Li obtained the first evidence of the distance-dependence of heterogeneous electron transfer rate constant for reduction of pentaaminecobalt(III) complexes anchored to gold and mercury surface. After Chidsey and co-workers reported heterogeneous electron transfer rate and electron tunneling constant (β) for the ferrocene-terminated alkanethiol monolayers with different alkyl chain length, a number of groups have reported β values for SAMs containing redox couples such as pentaamine (pyridine) Ru(II) complex, Os(II) bipyridyl complex, viologen, naphtoquinone, azobenzene, and hydroquinone. All of these reports demonstrated that the logarithmic heterogeneous apparent rate constant (kapp) linearly decreases as the distance between the electroactive center and the electrode surface increases as expected by Marcus theory. Those investigations on the distance dependence of electron transfer gave β values which are roughly ranging from 0.7 to 1.3 per methylene unit in the alkyl chain spacer. It is interesting to note that these β values are quite similar each other in their magnitude even though the electroactive SAMs contain different redox molecules and they follow significantly different electron transfer mechanism each other. For example, the electron transfer of ruthenium complex containing pentaamine and pyridine tether is highly reversible due to one-electron outer-sphere redox couple. However, redox center such as azobenzene undergoes substantially slow heterogeneous electron transfer because of its protonation reaction and structural change during its 2e−, 2H redox process. Among these redox centers studied, the electron transfer kinetics of a hydroquinone (H2Q) is also quite complicated due to its 2e −, 2H transfer reaction though Laviron presented a theoretical treatment of proton-coupled electron transfer reaction based on the nine-member square scheme. The electrochemical properties of hydroquinone/benzoquinone derivatives have been extensively studied in solutions because of their important biological activities. Especially, Hubbard and Soriaga have studied on the orientation of various quinones and mercaptohydroquinone derivatives adsorbed on metal surfaces using thin-layer electrochemistry. And Uosaki and coworkers have investigated the pH dependent redox behaviors of mercaptohydroquinone adsorbed on gold surface. We have reported the distance dependence of heterogeneous proton-coupled electron transfer rate constant of H2Q redox center in ω-mercaptoalkylhydroquinone SAMs on gold in 0.1 M HClO4 solution a few years ago. The β value reported for the H2Q-SAMs was 1.04 ± 0.06 and it was in good agreement with the values for the electroactive SAM systems reported up to now. However, the electron transfer kinetics of the H2Q-SAMs on gold shows remarkably different kinetic behavior in 0.1 M NaOH solution. In this note, we report an unusual observation for long range electron transfer in the ω-mercaptoalkylhydroquinone SAMs on gold electrode in a strong basic media. At this moment, the distance dependence of electron transfer of H2Q group is not observed in basic solution unlike that in acidic condition. Almost zero value of β determined from this phenomenon, to the best of our knowledge, is the first one in investigation of electron tunneling constant for long range electron transfer in the electroactive SAMs so far. The present work provides some understandings on the controlling factors related to proton-coupled electron transfer reaction nature.

This paper illustrates a TI N-Spire .tns file created by the author for generating continued fraction representations of real numbers and doing arithmetic with them. The continued fraction representation provides an alternative to the decimal representation. The .tns file can be used as tool for studying continued fractions and their properties as well as introducing the ideas of a best rational approximation to a real number and exploring some elementary ideas in error analysis. While the paper is a review of some of the classical works on continued factions, it does not present the material in a theorem/proof style. It motivates the results by citing examples and noting some interesting properties that will need to be verified. Notations are given to the references where exact and complete proofs of these facts can be found. The type of investigations done in this paper are motivational and the style of presentation is designed to pique students and teachers curiosity about the subject. Another purpose of this paper is to show that by looking at alternative representations of something that seems to be well known, such as the real numbers, can lead to exciting new and useful outcomes and applications.

It was shown recently that the anomalous scaling of simultaneous correlation functions in turbulence is intimately related to the breaking of temporal scale invariance, which is equivalent to the appearance of infinitely many times scales in the time dependence of time-correlation functions. In this paper we derive a continued fraction representation of turbulent time correlation functions which is exact and in which the multiplicity of time scales is explicit. We demonstrate that this form yields precisely the same scaling laws for time derivatives and time integrals as the "multi-fractal" representation that was used before. Truncating the continued fraction representation yields the "best" estimates of time correlation functions if the given information is limited to the scaling exponents of the simultaneous correlation functions up to a certain, finite order. It is worth noting that the derivation of a continued fraction representation obtained here for a time evolution operator which is not Hermitian or anti-Hermitian may be of independent interest.

We investigate thermodynamic and mechanical properties of silicon nanostructures at finite temperature. Thermodynamic properties for finite-temperature solid systems under isothermal conditions are characterized by the Helmholtz free energy density. The static part of the Helmholtz free energy is obtained directly from the interatomic potential, while the vibrational part is calculated by using the theory of local phonon density of states (LPDOS). The LPDOS is calculated efficiently from the on-site phonon Green's function by using a recursion technique based on a continued fraction representation. The Cauchy-Born hypothesis is employed to compute the mechanical properties. By considering ideal Si{001}, $(2\ifmmode\times\else\texttimes\fi{}1)$ reconstructed Si{001}, and monolayer-hydrogen-passivated $(2\ifmmode\times\else\texttimes\fi{}1)$ reconstructed Si{001} surfaces of a silicon nanowire, we calculate the local phonon structure and local thermodynamic and mechanical properties at finite temperature and observe that the surface effects on the local thermal and mechanical properties are localized to within one or two atomic layers of the silicon nanowire.

A Green's function theory applicable to magnetic systems with complicated levels for each ion is developed using the standard basis operators. A diagrammatic technique is employed. This provides a self-consistent approach to the rare-earth and actinide systems where the crystal field is of the same order as the exchange interaction. A Wick-like theorem is shown first followed by a discussion of the diagrammatic representation. Guided by a priority rule we consider all possible vertices. Rules for construction and evaluation of diagrams are given and summation of diagrams is discussed.

4 - Continued Fractions and Indeterminate Equations

Electronic densities of states of tight-binding s-bands for simple cubic and body-centred cubic lattices were determined using the modified-moments method. The weight functions close to the density-of-states functions for SC and BCC lattices were found. These functions define the orthogonal polynomials belonging to the class of polynomials considered by Bernstein and Szego. The analytical expressions for modified moments were derived and the densities of states in the form of continued fractions which reproduce 76 moments for the SC and 78 moments for the BCC lattice were obtained and compared with exact results. The convergence of the modified-moments method as a function of the level at which the continued fraction is terminated in an approximate way has been examined.

Extrapolation algorithm for the terminator problem in the recursion method

Abstract—This review paper reports the data on the role of long range electron tunneling in photosynthetic and model systems. The main papers concerned with elucidation mechanisms and kinetical peculiarities of electron transfer in reaction centers of bacteria and green plants are considered. The paper reviews the articles on long range electron transfer in reactions of metalloporphyrins—synthetic analogs of the main natural pigment of photosynthesis, chlorophyll. Liquid and solid phase redox reactions of electronically excited porphyrin molecules, intramolecular and photosensitized electron transfer processes are considered.

Long range electron transfer (LRET) across protein matrix underlies all one-electron cellular redox reactions. Elucidation of molecular electron transfer pathways and parametrization of their relative efficiency is one of the most challenging problems in the studies on LRET in proteins. In this paper results of pulse radiolysis investigations on kinetics of LRET accompanying intramolecular radical transformation Trp. --> TyrO. in model peptides built of tryptophan and tyrosine bridged by an oligoproline fragment are reviewed, along with an interpretation of the observed distance dependence of the rate of LRET in terms of conformational properties of the peptides, and partitioning of LRET between electron transfer pathways through space and through peptide backbone. This review on model peptide systems is supplemented with recapitulation of similar studies on the same intramolecular transformation in hen egg-white lysozyme, which allowed to identify Trp./Tyr redox pairs and associated electron transfer pathways involved in LRET in this protein.

Starting with the Liouville equation and with use of the combined projection technique presented earlier [J. Y. Sug, C. H. Choi, and S. D. Choi, Nuovo Cimento B 109, 177 (1994)], we obtain the ensemble average of current of many-electron systems in the presence of oscillatory electric fields. For quite weak fields the dynamic conductivity is given in a continued fraction form. The present form contains the first- and second-order correction parts, while the Mori type contains only the first-order part.

This is a bibliographical review that provides a brief glimpse into the beauty of continued fractions and how they can be very useful, both in obtaining good rational approximations of a given real number and in their representation as a continued fraction. Furthermore, it presents important properties, such as the relationship between quadratic irrationals and periodic continued fractions. On the other hand, with a view to a potential introduction of this topic in elementary education, a method for obtaining square root approximations through continued fractions is presented. Finally, using more advanced tools, we present an infinite continued fraction representation of the Euler number, which consequently implies the irrationality of e.

A thorough exposition of finite simple continued fractions is given. All rational numbers have such representations. Convergents to a simple continued fraction is studied and a handy algorithm is developed which helps to readily compute them. The uniqueness of a continued fraction representation is covered. The theory is naturally extended to infinite simple continued fractions including those for $\sqrt 2 $ and for $\pi $ as well as the general importance of finite continued fractions as approximations to real numbers. The chapter culminates in a robust discussion of Christiaan Huygens’ cogwheeled planetarium which utilized continued fractions.

Kinetics and nonlinear optical properties of a long range electron transfer complex in a polar solvent are studied in the presence of two perpendicularly directed, independent (incoherent) lasers with electric fields aligned along the electron path. The rate constant and absorption/emission cross sections in both direction are computed on the basis of the channel approach, whose validity is justified by a rigorous derivation. It is found that despite strong dissipation due to interaction with the polar environment, emission can be stimulated in one (“primary”) direction. The emission spectra are found to depend on the frequency of the auxiliary laser propagating perpendicular to the direction of the primary laser, the intensities of both lasers, the frequency and reorganization energy of quantum modes in the solvent, and the reaction heat. The emission spectrum is only slightly sensitive to the laser intensities in certain high frequency regimes. The latter property is important for potential applications in which the reaction system would be employed as the working system for a laser with tunable frequency. Calculations suggest that a long range electron transfer system can be also used as a nonlinear optical device which transfers energy from one laser to another.

LetFn be the set of fractions p=q2 [0; 1] such that the sum of partial quotients in their continued fraction representation is not greater than n. We consider the sum (Fn) = P (qq 0 ) taken over the denominators of neighbouring fractions inFn and prove that for all > 1 (Fn) = 2 n