Golden Rule Approach Research Articles

Fermi Golden Rule Approach to Evaluating Outer-Sphere Electron-Transfer Rate Constants at Semiconductor/Liquid Interfaces

Fermi's golden rule is used to formulate rate expressions for charge transfer of delocalized carriers in a nondegenerately doped semiconducting electrode to localized, outer-sphere redox acceptors in an electrolyte phase. If the charge-transfer rate constant is known experimentally, these rate expressions allow computation of the value of the electronic coupling matrix element between the semiconducting electrode and the redox species. This treatment also facilitates comparison between charge-transfer kinetic data at metallic and semiconducting electrodes in terms of parameters such as the electronic coupling to the electrode, the attenuation of coupling with distance into the electrolyte, and the reorganization energy of the charge-transfer event. Within this framework, rate constant values expected at representative semiconducting electrodes have been evaluated from experimental data for charge transfer from Au electrodes to ferrocene-terminated thiols, to Ru(NH3)53+/2+-terminated thiols, and through bl...

X-ray absorption problem in metals within the one-electron approximation: The temperature dependence

The golden-rule approach to the X-ray-absorption problem in metals based on the one-electron approximation derived in previous work (V.M. Adamjan, J. Ortner et al. Phys. Rev. B 52 (1995) 13 827) is generalized to finite temperature. A general expression for the X-ray absorption intensity in terms of invariants of one-particle operators is given. The analytic form of the spectra in the near-edge region is derived.

Transport properties of a quantum dot with superconducting leads.

We report a numerical study of transport properties of a quantum dot with superconducting leads. We introduce a general phenomenological model of quantum dot transport, in which electron tunnel rates are computed within the Fermi's Golden Rule approach. The low temperature current-voltage (I-V) characteristics are in qualitative agreement with experimental observations of Ralph et al. [Phys. Rev. Lett., 74, 3241 (1995)]. At higher temperatures, our results reveal new effects due to the thermal excitation of quasiparticles in the leads as well as the thermal population of excited quantum levels in the dot. We also study the photon-assisted tunneling phenomena in our system and point out its potential for millimeter wave applications.

Dissipative tunneling in asymmetric double-quantum-well systems: A coherence phenomenon.

Recent experiments on asymmetric double-quantum-well systems revealed unexpected long time scales of resonant tunnel processes as well as deviations from the expected dependence of the tunneling lifetime T on the barrier thickness. Both effects are related to the presence of dissipative processes. Starting from microscopic expressions for T beyond the usual golden rule approach we show that the observed behavior is a consequence of Fano interference, which implies that low-temperature dissipative tunneling can be treated as a totally coherent process. Including disorder due to interface roughness, these Fano interferences lead to a nonexponential and slowed-down decay. \textcopyright{} 1996 The American Physical Society.

A three-dimensional wave packet study of Ar⋅ ⋅ ⋅I2(B )→Ar + I + I electronic predissociation

A three-dimensional wave packet study of Ar...I2(B)→ Ar + I(2P3/2)+ I(2P3/2) electronic predissociation, arising from the argon-induced electrostatic coupling between the B(3Π0+u) and the repulsive a(3Π1g) state of I2, is presented. A time-dependent golden rule approach is used. The initial wave packet corresponds to a bound vibrational wave function of the Ar...I2(B) complex (with zero total angular momentum) multiplied by the electronic coupling. A 3-D propagation in the final dissociative surface is then performed and the predissociation rates are obtained by Fourier transform of the wave packet autocorrelation function. The potential energy surfaces are described by sums of atom–atom interactions. For the B(3Π0+u) state potential, empirically determined van der Waals parameters available from the literature are used. For the final dissociative a(3Π1g) electronic state, the van der Waals parameters are adjusted to reproduce the experimentally observed oscillations of the electronic predissociation rate as a function of the initial vibrational quantum number v′ of I2. It is shown that good agreement between calculated and measured values can be obtained with a van der Waals well of 100 cm−1 and an interstate coupling of the order of 14 cm−1.

X-ray-absorption problem in metals within the one-electron approximation.

An alternative golden-rule approach to the calculation of x-ray intensity in metals based on the one-electron approximation is proposed. The model permits us to calculate the critical exponent of the x-ray-edge power-law behavior without adjustable parameters, such as the contact core-hole potential intensity and is in good agreement with available experimental data.

Multimode approach to hydrogen tunneling

A procedure is developed to improve the quantitative evaluation of hydrogen transfer rates in polyatomic molecules and solids. The aim is to introduce a dynamical model that includes explicitly all vibrations participating in the transfer. This aim favors adoption of the golden-rule approach, since it treats all vibrational modes equally. To simplify the resulting multidimensional transfer integrals, two basic assumptions are introduced: (i) adiabatic separability of the hydrogenic modes directly involved in the tunneling from the other modes, and (ii) negligible anharmonicity for the latter. The number of effectively participating modes can then be reduced drastically by transformation to an appropriate local representation which allows analytical integration over most of these other modes. Those that remain involve vibrations of the atoms between which the hydrogen is transferred. Their frequency, reduced mass and displacement are expressed in terms of the harmonic force field of the system before and after transfer and can be unambiguously evaluated if these force fields are available. These modes replace the empirical effective modes used previously. The theory is applied successfully to single hydrogen transfer in dimethyl-glyoxime and double hydrogen transfer in porphine.

Generalized golden rule approach to the H-transfer problem. II. Application to a photochemical reaction

The golden rule approach generalized for arbitrary values of the electronic coupling integral, as reported earlier, is applied to interpret the kinetic data and discuss the driving forces of the fast photochemical H- and D-transfer in acridine-doped fluorene crystal. The excited-state proton transfer (ESPT) is treated as multidimensional tunneling. Both the stretching and bending vibrations of the transferred H atom and the lattice phonons which affect the tunneling distance are taken into account. The relatively large value of the electronic coupling J between the initial and final state corresponding to the fluorene-triplet acridine, and triplet acridinyl-fluorenyl radical pair complex, respectively, is found to be the main reason for the unusually rapid reaction. This conclusion is supported by a direct estimate of J using Slater orbitals.

Direct estimate of the electronic coupling driving the photochemical proton transfer in acridine-doped fluorene crystal

The electronic coupling integral J of a photochemical hydrogen abstraction in an acridine-doped fluorene crystal was estimated in order to clarify the controversial question of its driving force. The features of the reaction sytem strongly motivate the adoption of the “diabatic” procedure for calculating J instead of its calculation as an adiabatic quantity. A simple model was applied for direct construction of the diabatic states employing a single determinant presentation of the wavefunctions. AO s being Slater orbitals. The present estimates support the previously made conclusion from the analysis of the kinetic data based on a golden-rule approach generalized for arbitrary values of the coupling, that J is mainly responsible for the unusually rapid reaction.

Generalized golden rule treatment of a photochemical solid-state reaction involving hydrogen tunneling

Fast photochemical H- and D-transfer in acridine doped fluorene crystals is treated as multidimensional tunneling. Both the stretching and bending vibrations of the transferred atom and the lattice phonons which affect the tunneling distance are taken into account. The golden rule approach is applied as generalized for arbitrary values of the electronic coupling integral, as reported earlier. Its relatively large value is found to the main reason for the unusually rapid reaction.

Golden-rule treatment of hydrogen abstraction by photoexcited acridine guests in fluorene single crystals

The kinetics of the photoreduction of acridine guest molecules in fluorene single crystals has been investigated theoretically. The abstraction has been modeled using the golden-rule approach and proceeds by vibrationally assisted tunneling. Parameter values have been taken from experimental data and from semiempirical force-field calculations. The calculated temperature and isotope effects are in good agreement with the experimental data for acridine and phenazine guests [B. Prass, J. P. Colpa, and D. Stehlik, J. Chem. Phys. 88, 191 (1988)]. While a fluorene lattice vibration is the likely promoting mode, NH wagging in the radical-pair product plays an essential role.

Generalized golden rule approach to the H-transfer problem in polyatomic systems. I. Multimode nuclear effects in solids

Golden rule (GR) analysis is reported of the multimode nuclear effects on intermolecular H (D) tunneling. The collinear reaction model is applied with realistic quadratic expansion of the hydrogen tunneling factor over normal coordinate displacements. For an arbitrary multimode system, a rate constant expression is derived valid for the whole temperature region and convenient for numerical analysis. The validity of this, as well as of the linear expansion is quantitatively discussed. It is demonstratedthat the multimode problem for intermolecular tunneling in solids with Debye phonon spectrum can be treated with the explicit form of the tunneling factor by introducing an effective local mode. As an example of a multimode system, radical-pair conversion in crystalline dimethylglyoxime is discussed in detail for both isotopes. The available kinetic data are fitted within a GR approach generalized for an arbitrary value of the coupling as reported earlier.

Golden-rule approach to the soft-x-ray-absorption problem. V. Thermal broadening and comparison with experiments in quantum wells.

Exact spectra of photon emission and absorption are given for the Mahan--Nozi\`eres--De Dominicis (MND) Hamiltonian in the finite-temperature case. Besides infrared divergence, existence of a bound state and relaxation shift of the threshold energy are rigorously taken into account. We discuss in detail the interplay between the infrared divergence and thermal broadening and examine how the results in the ladder approximation are improved by the exact treatment. Based on the numerical results, quantitative comparison with recent experiments for quantum wells with ${\ensuremath{\varepsilon}}_{F}$=200 K and 4.2 KlTl60 K is attempted. In the case of absorption it is found that if an appropriate value for the lifetime of a core hole is assumed, the MND Hamiltonian is capable of yielding a line shape in fairly good agreement with the experiments. For the emission spectra, however, the agreement is worse and it seems that some dissipative mechanism operating prior to photon emission has to be included.

Edge Singularity and Fano Effect in Metals

The X-ray absorption and emission in metals with a localized (resonant) state within the band are treated both analytically and numerically. The model introduced by Kotani and Toyozawa is generalized so as to take into account the scattering of the conduction electrons by the core-hole potential and analyzed by the Fermi golden rule approach. Exact analytic expressions are derived for the line shapes near the threshold and near the localized state while the spectra over the entire frequency range are obtained numerically within an error of 1%. The competition between the edge singularity, the Fano effect and the peak corresponding to the localized state give rise to a variety of the spectral line shapes. In some cases disappearance of the edge singularity due to the Fano effect is shown to be observed.

Golden-rule approach to the soft-x-ray-absorption problem. II. The effect of a bound state

The soft-x-ray-absorption problem is examined in the case where a bound state exists in the final state. The analysis is based on the Fermi-golden-rule approach developed in a previous paper [Phys. Rev. B 28, 6833 (1983)]. It is found that the present method leads quite naturally to the conclusions of Combescot and Nozi\`eres (CN), who first examined this problem. Besides reproducing the results of CN, the present analysis gives exact expressions of the integrated intensities of the optical absorption for both absolute and secondary absorption bands. The formulas reveal that there should be a marked difference between the x-ray-photoelectron spectrum (XPS) and optical-absorption spectrum in the intensity of the secondary band. Also, the present work leads to expressions for the critical amplitudes, the prefactors of the power-law spectra, of the main and secondary thresholds, which are exact for a contact-type core-hole potential. For these two quantities some numerical results are presented for a model conduction band.

Golden-rule approach to the soft-x-ray-absorption problem

The cross section of the soft-x-ray absorption is derived based on the Fermi golden rule, by summing the transition probabilities over all possible final states. It is shown that the factor connected with the Anderson orthogonality catastrophe is canceled in the process of summation, and the spectrum is finally expressed by the core-hole propagator times the open-line propagator, as shown by Nozi\`eres and De Dominicis. Our results are compared with the results obtained by Mahan and collaborators and are shown to improve over theirs in several respects, expecially in the core-hole propagator. With the use of the new integral kernel, the integral equation for the threshold region is treated and the exact form of the edge anomaly, including the analytical form of the prefactor of the power-law spectrum of Nozi\`eres and De Dominicis, is obtained.

Price and quantity adjustment in a dynamic closed model: The dual stability theorem

The paper analyzes dual instability in a dynamic input-output framework. It is divided into three parts: the first gives an informal introduction, the second discusses the similarity with the knife edge, and the last suggests a new interpretation, which guarantees nonnegative solutions and relative stability. It has been assumed that the producers adjust their prices following an adaptive expectation mechanism, and the current level of investment to the expected value defined by a golden rule approach.