We present a three-dimensional model system which allows for a detailed investigation of the superexchange coupling (VS) between distant redox centers in large electron-transfer systems. The system (bridge) is described by a tight-binding Hamiltonian for an aperiodic array of sites interacting through an exponentially decaying function of distance. The dependence of VS on the donor-acceptor distance, the density of sites in the medium, and the energy separation between the bridge and the redox sites is studied numerically. We present an analytical calculation, based on a continuous-medium approximation, which accounts for most of the numerical results. Evidence is given for the existence of a unique control parameter (EIT), expressed by the ratio of the energy gap to the half-bandwidth of the medium. Due to interferences between the large number of pathways, it is found that the sign of (EIT), which distinguishes electron from hole transfers, is of particular importance. Depending on this sign, the effective coupling differs by many orders of magnitude and, while presenting a monotonic variation for the case EIT 0, shows an oscillatory behavior for the case EIT > 0. These results are beyond the usual McConnell or WKB approximations.
Monte Carlo simulations of the radiolysis of neutral liquid water and 0.4 M H(2)SO(4) aqueous solutions at ambient temperature are used to calculate the variations of the primary radical and molecular yields (at 10(-6)s) as a function of linear energy transfer (LET) in the range approximately 0.3 to 6.5 keV/micrometer. The early energy deposition is approximated by considering short (approximately 20-100 micrometer) high-energy (approximately 300-6.6 MeV) proton track segments, over which the LET remains essentially constant. The subsequent nonhomogeneous chemical evolution of the reactive species formed in these tracks is simulated by using the independent reaction times approximation, which has previously been used successfully to model the radiolysis of water under various conditions. The results obtained are in good general agreement with available experimental data over the whole LET range studied. After normalization of our computed yields relative to the standard radical and molecular yields for (60)Co gamma radiation (average LET approximately 0.3 keV/micrometer), we obtain empirical relationships of the primary radiolytic yields as a function of LET over the LET range studied. Such relationships are of practical interest since they allow us to predict a priori values of the radical and molecular yields for any radiation from the knowledge of the average LET of this radiation only. As an application, we determine the corresponding yields for the case of (137)Cs gamma radiation. For this purpose, we use the value of approximately 0.91 keV/micrometer for the average LET of (137)Cs gamma rays, chosen so that our calculated yield G(Fe(3+)) for ferrous-ion oxidation in air-saturated 0.4 M sulfuric acid reproduces the value of 15.3 molecules/100 eV for this radiation recommended by the International Commission on Radiation Units and Measurements. The uncertainty range on those primary radical and molecular yields are also determined knowing the experimental error (approximately 2%) for the measured G(Fe(3+)) value. The following values (expressed in molecules/100 eV) are obtained: (1) for neutral water: G(e(-)(aq)) = 2.50 +/- 0.16, G(H(.)) = 0.621 +/- 0.019, G(H(2)) = 0.474 +/- 0.025, G((.)OH) = 2.67 +/- 0.14, G(H(2)O(2)) = 0.713 +/- 0.031, and G(-H(2)O) = 4.08 +/- 0.22; and (2) for 0.4 M H(2)SO(4) aqueous solutions: G(H(.)) = 3.61 +/- 0.09, G(H(2)) = 0.420 +/- 0.019, G((.)OH) = 2.78 +/- 0.12, G(H(2)O(2)) = 0.839 +/- 0.037, and G(-H(2)O) = 4.46 +/- 0.16. These computed values are found to differ from the standard yields for (60)Co gamma rays by up to approximately 6%.
Non-nearest-neighbor interactions in the problem of electron transfer via superexchange are considered. In particular, second-neighbor coupling in a finite disordered chain and exponential coupling in an infinite periodic chain are investigated in detail. Inclusion of these interactions is formulated through a renormalization of the energies as well as of the coupling matrix elements which are constructed by a simple recursive scheme. The form of the extended McConnell formula is then applicable. It is shown that, under certain conditions, the effective coupling between the donor and acceptor sites may suffer large changes.Extensive theoretical work has recently been devoted to the understanding of the role of the intermediate medium in the determination of the electronic coupling TAD between distant redox centers.l This coupling is a major factor involved in the rate of electron-transfer reactions2 occurring in a wide variety of systems, ranging from small chemical to huge biological molecules. For nonadiabatic reactions, TAD is related to the electron-transfer rate constant k via the generalized version of Fermi's Golden Rulewhere h is Planck's constant divided by 27 and (FCWD) is the so-called Franck-Condon-weighted density of states which accounts for the contribution of nuclear coordinates to the kinetic process. The total electronic coupling TAD between the acceptor (A) and donor (D) sites is the sum of the direct coupling VAD and the elecyonic superexchange coupling VS which is the quantity of interest in this work.The simple one-electron, tight-binding Hamiltonian is used as the starting point of most theoretical calculations. So far, a few analytical results have been obtained for the case of nearestneighbor interactions, starting with the work of McConnell,' who gave the expression for a one-dimensional bridge consisting of N identical units. In eq 2, Tis the coupling of A and D to the bridge, t is the coupling between two adjacent units in the bridge, and E is the energy gap between the redox sites and the bridge. McConnell's formula has then been generalized to the case of aperiodic bridging chains presenting both diagonal and off-diagonal disorder so that1 (3)where tit+l is a nearest-neighbor coupling matrix element of the chain, Ei is the energy gap between the bridge site i and the redox *Abstract published in Aduance ACS Absrracrs, August 15, 1993. 0022-3654/93/2097-9266$04.00/0 sites, and VD and VA denote the couplings of D and A to the chain ends, respectively. Equations 2 and 3 give the contribution of one single electron-transfer pathway to VS. Within the nearestneighbor approximation, it has been shown that, for situations in which multiple bridge pathways can occur between donor and acceptor, the exact expression of V. assumes the same form as in eq 3, where the single-pathway energy parameters Ei are now replaced by a renormalized expre~sion.~ From recent work it is becoming clear that inclusion of nonnearest-neighbor interactions, and consequently of many interfering pathways, is necessary...
A re-examination of our Monte-Carlo modeling of the radiolysis of liquid water by low linear-energy-transfer (LET ~ 0.3 keV µm1) radiation is undertaken herein in an attempt to reconcile the results of our simulation code with recently revised experimental hydrated electron (eaq) yield data at early times. The thermalization distance of subexcitation electrons, the recombination cross section of the electrons with their water parent cations prior to thermalization, and the branching ratios of the different competing mechanisms in the dissociative decay of vibrationally excited states of water molecules were taken as adjustable parameters in our simulations. Using a global-fit procedure, we have been unable to find a set of values for those parameters to simultaneously reproduce (i) the revised eaq yield of 4.0 ± 0.2 molecules per 100 eV at "time zero" (that is, a reduction of ~20% over the hitherto accepted value of 4.8 molecules per 100 eV), (ii) the newly measured eaq decay kinetic profile from 100 ps to 10 ns, and (iii) the time-dependent yields of the other radiolytic species H, OH, H2, and H2O2 (up to ~1 µs). The lowest possible limiting "time-zero" yield of eaq that we could in fact obtain, while ensuring an acceptable agreement between all computed and experimental yields, was ~4.4 to 4.5 molecules per 100 eV. Under these conditions, the mean values of the electron thermalization distance and of the geminate electroncation recombination probability, averaged over the subexcitation electron "entry spectrum," are found to be equal to ~139 Å and ~18%, respectively. These values are to be compared with those obtained in our previous simulations of liquid water radiolysis, namely ~88 Å and ~5.5%, respectively. Our average electron thermalization distance is also to be compared with the typical size (~6480 Å) of the initial hydrated electron distributions estimated in current deterministic models of "spur" chemistry. Finally, our average probability for geminate electroncation recombination agrees well with an estimated value of ~15% recently reported in the literature. In conclusion, this work shows that an adaptation of our calculations to a lower hydrated electron yield at early times is possible, but also suggests that the topic is not closed. Further measurements of the eaq yields at very short times are needed. Key words: liquid water, radiolysis, electroncation geminate recombination, electron thermalization distance, hydrated electron (eaq), eaq decay kinetics, time-dependent molecular and radical yields, Monte-Carlo simulations.
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