Electron transfer quantum mechanical theory

Despite the fact that Bohr s stopping power theory is useful for heavy charged particles such as fission fragments, Rutherford s collision cross section on which it is based is not accurate unless both the incident particle velocity and that of the ejected electron are much greater than that of the atomic electrons. The quantum mechanical theory of Bethe, with energy and momentum transfers as kinematic variables, is based on the first Born approximation and certain other approximations [1,2]. This theory also requires high incident velocity. At relatively moderate velocities certain modifications, shell corrections, can be made to extend the validity of the approximation. Other corrections for relativistic effects and polarization screening (density effects) are easily made. Nevertheless, the Bethe-Born approximation... 

Instead of the quantity given by Eq. (15), the quantity given by Eq. (10) was treated as the activation energy of the process in the earlier papers on the quantum mechanical theory of electron transfer reactions. This difference between the results of the quantum mechanical theory of radiationless transitions and those obtained by the methods of nonequilibrium thermodynamics has also been noted in Ref. 9. The results of the quantum mechanical theory were obtained in the harmonic oscillator model, and Eqs. (9) and (10) are valid only if the vibrations of the oscillators are classical and their frequencies are unchanged in the course of the electron transition (i.e., (o k = w[). It might seem that, in this case, the energy of the transition and the free energy of the transition are equal to each other. However, we have to remember that for the solvent, the oscillators are the effective ones and the parameters of the system Hamiltonian related to the dielectric properties of the medium depend on the temperature. Therefore, the problem of the relationship between the results obtained by the two methods mentioned above deserves to be discussed. 

Both the initial- and the final-state wavefunctions are stationary solutions of their respective Hamiltonians. A transition between these states must be effected by a perturbation, an interaction that is not accounted for in these Hamiltonians. In our case this is the electronic interaction between the reactant and the electrode. We assume that this interaction is so small that the transition probability can be calculated from first-order perturbation theory. This limits our treatment to nonadiabatic reactions, which is a severe restriction. At present there is no satisfactory, fully quantum-mechanical theory for adiabatic electrochemical electron-transfer reactions. 

A well defined theory of chemical reactions is required before analyzing solvent effects on this special type of solute. The transition state theory has had an enormous influence in the development of modern chemistry [32-37]. Quantum mechanical theories that go beyond the classical statistical mechanics theory of absolute rate have been developed by several authors [36,38,39], However, there are still compelling motivations to formulate an alternate approach to the quantum theory that goes beyond a theory of reaction rates. In this paper, a particular theory of chemical reactions is elaborated. In this theoretical scheme, solvent effects at the thermodynamic and quantum mechanical level can be treated with a fair degree of generality. The theory can be related to modern versions of the Marcus theory of electron transfer [19,40,41] but there is no... 

Part A. Quantum-Mechanical Theory of Diffusion Independent Electron Transfer in Biological Systems by Ephraim Buhks (University of Delaware)... 

The molecular concept has become so central in chemistry that understanding of chemical events is commonly assumed to consist of relating experimental observations to micro events at the molecular level, which means changes in molecular structure. In this sense molecular structure is a fundamental theoretical concept in chemistry. As the micro changes are invariably triggered by electron transfer, the correct theory at the molecular level must be quantum mechanics. It is therefore surprising that a quantum theory of molecular structure has never developed. This failure stems from the fact that physics and chemistry operate at different levels and that grafting the models of physics onto chemistry produces an incomplete picture. 

Nov. 21, 1931, Tbilisi, Georgia, USSR - May 13, 1985) Dogonadze was one of the founders of the new science - electrochemical physics [i]. The main scientific interests of Dogonadze were focused on condensed-phase reactions. His pioneering works of 1958-59 have laid the foundations of the modern quantum-mechanical theory of elementary chemical processes in electrolyte solutions. He developed a comprehensive quantum-mechanical theory of the elementary act of electrochemical reactions of -> electron and -> proton transfer at metal and - semiconductor electrodes [ii—v]. He was the first to obtain, by a quantum-mechanical calculation, the expression for the electron transfer probability, which was published in 1959 in his work with -> Levich. He conducted a number of studies on the theory of low-velocity electrons in disordered systems, theory of solvated electrons, and theory of photochemical processes in solutions. He made an impressive contribution to the theory of elementary biochemical processes [vi]. His work in this area has led to the foundation of the theory of low-temperature -> charge-transfer processes cov-... 

In the volumes to come, special attention will be devoted to the following subjects the quantum theory of closed states, particularly the electronic structure of atoms, molecules, and crystals the quantum theory of scattering states, dealing also with the theory of chemical reactions the quantum theory of time-dependent phenomena, including the problem of electron transfer and radiation theory molecular dynamics statistical mechanics and general quantum statistics condensed matter theory in general quantum biochemistry and quantum pharmacology the theory of numerical analysis and computational techniques. 

Levich, V.G. and Dogonadze, R. ( 1959) Quantum mechanical theory of electron transfer in polar media. Dokl. 

Recently Mulliken 5 has given a more general quantum mechanical theory of the bonding in these molecular complexes. He also considers resonance between no-bond configurations A, B with dative configurations A" B+. These latter ones are, however, equivalent to the configurations of Brack-man, which also arise from electron transfer from the N-com-pound or donor (Lewis base) to the c sextet5 -compound or acceptor (Lewis acid, p. 87). 

Because of the inadequacies of QED a fundamental theory of electrode processes is still lacking. The working theories are exclusively phenomenological and formulated entirely in terms of ionic distributions in the vicinity of electrode interfaces. An early, incomplete attempt [54] to develop a quantum mechanical theory of electrolysis based on electron tunnelling, is still invoked and extensively misunderstood as the basis of charge-transfer. It is clear from too many superficial statements about the nature of electrons that the symbol e is considered sufficient to summarize their important function. The size, spin and mass of the electron never feature in the dynamics of electrochemistry. 

In the quantum-mechanical theories the intersection of the potential energy surfaces is deemphasized and the electron transfer is treated as a radiationless transition between the reactant and product state. Time dependent perturbation theory is used and the restrictions on the nuclear configurations for electron transfer are measured by the square of the overlap of the vibrational wave functions of the reactants and products, i.e. by the Franck-Condon factors for the transition. Classical and quantum mechanical description converge at higher temperature96. At lower temperature the latter theory predicts higher rates than the former as nuclear tunneling is taken into account. 

No further chemical input is necessary for substrate oxidation to occur once the second electron is transferred. Quantum mechanical calculations have provided insights into the protonation and 0-0 bond cleavage steps. Density function theory (DFT) calculations on the ferric-peroxo form of P450eryF indicated fast protonation at the distal oxygen by... 

Predictions of the kinetics of electrons taking into account all size-dependent factors are possible only when adeqnate ion-molecular models of reaction layers are bnilt. For a number of systems, this problem can be solved snccessfully by employing qnantum-chemical methods based on quantum mechanical theory of the charge-transfer elementary act [74,75] along with the classical effects of the cation size, which are manifested in the rednction of anions on a negatively charged snrface [74,75]. 

The transmission coefficient k which is also included in the expression for the preexponential factor, cannot be calculated using a purely phenomenological approach, but involves the quantum mechanical theory of an elementary act. For most simple electron transfer reactions, k = 1. 

Let us consider the simple case of two ions, each with one excitable electronic state separated from its electronic ground state by nearly equal energy. With suitable interaction between the two electronic systems, the excitation will jump from one ion to the other before a quantum of fluorescence is emitted. The systems interact by Coulomb interactions of the Van der Waals type. Forster (1948), who first treated such a case by quantum-mechanical theory, considered the dipole-dipole interactions. He assumed that the interaction is strongest if, for both transitions, electric-dipole transitions are allowed (Forster 1960). The interaction energy (//sa) is then proportional to the inverse of the third power of the interionic distance, and the transfer probability is given by... 

Theoretically, Frumkin strongly supported quantum theoretical studies of the elementary act of electron transfer. In the Theoretical Division, led by V. G. Levich, several epochal studies were carried out by R. R. Dogonadze, Yu. A. Chizmadzhev, and A. M. Kuznetsov. In 1967-1968, Dogonadze, Kuznetsov, and Levich put forward the first quantum-mechanical theory of proton transfer later, A. M. Kuznetsov, J. Ulstrup, Ya. I. Kharkats, and M. A.Vorotyntsev systematically... 

Quantum-mechanical theory describes the behavior of electrons in atoms. Since chemical bonding involves the transfer or sharing of electrons, quantum-mechanical theory helps us understand and describe chemical behavior. As we saw in Chapter 7, electrons in atoms exist within orbitals. An electron configuration for an atom shows the particular orbitals that electrons occupy for that atom. For example, consider the ground state—or lowest energy state—electron configuration for a hydrogen atom ... 

Abstract This chapter introduces the supramolecular photochemistry, i.e. photochemistry applied to supramolecular systems, and discusses the thermodynamic and kinetic aspects of photoinduced energy and electron transfer processes both between molecules and within supramolecular systems. In the case of electron transfer processes, Marcus theory is presented as well as quantum mechanical theory. For energy transfer processes, coulombic and exchange mechanisms are illustrated and the role of the bridge in supramolecular structures is discussed. 

Soon after it was established that the discharge stage has a finite rate, efforts began to be made to develop a quantum-mechanical theory of an elementary act. The first important step in this direction was taken by Gurney[6]. He clearly formulated the basic idea that the transfer of an electron from a metal to an ion in solution (or in the reverse direction) can be accomplished only when the energy levels of the initial and the final states turn out to be virtually identical (for example, an electron on the Fermi level in a metal + a singly-charged ion in solution is the initial state and an electron + an ion, i.e. the atom in solution is the final state). 

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