Matrix elements electron-vibrational

Connection with vibrational lifetime on surfaces. The decay of molecular vibrations in the excitation of the electron-hole pairs of metallic surfaces have been identified with the mechanisms of vibration excitation by tunneling electrons [42]. Intuitively this may seem so. Indeed, an excited vibration may couple to the surface electronic excitations through the same electron-vibration matrix elements of Eqs. (2) and (4). The surface... 

Let us consider the electron-vibrational matrix element. As is usually done, we consider two coordinate systems, the origins of which are located at the center of mass of the molecule. The first coordinate system is fixed in space, while the second system (the rotational one) is fixed to the molecule. For describing the orientation of the rotational system with respect to the fixed frame we use the Eulerian angles 6 = a, / , y. In the Born-Oppenheimer approximation, the motion of nuclei may be decomposed into the vibrations of the nuclei about their equilibrium position and the rotation of the molecule as a whole. Accordingly, the wave function of the nuclei X (R) is presented as a product of the vibrational wave function A V(Q) and the rotational wave function... 

Substituting wave functions of Eq. (18) into the matrix element Eq. (16) and isolating the electron-vibrational matrix element we get... 

Let us single out the electron part of the electron-vibrational matrix element. To this end let us consider its integrand versus the distance between the nuclei R ... 

Adiabatic and Non-Condon. The non-Condon approaches, as mentioned in Section 10c, retain some interaction between the electronic and vibrational matrix elements. As a general conclusion, two primary results emerge from all these treatments. First, the temperature dependence of the transition probability is still mainly determined by the vibrational levels. This follows since the... 

From Eq. [239], it is apparent that the size of a particular is not only determined by the magnitude of the electronic coupling matrix element but also by the overlap of the vibrational wave functions v,- and i/. Squared overlap integrals of the type (Xi/, (Q) IXt/ (Q))q 2 are frequently called Franck-Con-don (FC) factors. In contrast to radiative processes, FC factors for nonradiative transitions become particularly unfavorable if two states differing considerably in their electronic energies exhibit similar shapes and equilibrium coordinates of their potential curves. Due to the near-degeneracy requirement, an upper state vibrational wave function, with just a few nodes... 

The following variables are used in Eqs. (1) and (2) HAB is the electronic coupling matrix element that permits ET to occur h is Planck s constant k is Boltzmann s constant T is absolute temperature s is the reorganization energy of the solvent vibrations associated with ET wy is the angular frequency of the quantized, high-energy molecular vibration associated with ET, such that ooy/27r = m in the... 

The Fermi Golden rule describes the first-order rate constant for the electron transfer process, according to equation (11), where the summation is over all the vibrational substates of the initial state i, weighted according to their probability Pi, times the square of the electron transfer matrix element in brackets. The delta function ensures conservation of energy, in that only initial and final states of the same energy contribute to the observed rate. This treatment assumes a weak coupling between D and A, also known as the nonadiabatic limit. 

Each term of equations (6)-(8) is factorized into parts that can be related to the electronic and vibrational wavefrmctions and excitation frequency. The electronic contribution (matrix elements of /x° and /u. ), weighted with the differential energy denominators — vo + Tu). determines the total enhancement of all the normal vibrations for a given... 

The first term of the right hand side of Eq. (7) represents the squared electronic dipole matrix element and specifies the intensity of the purely electronic transition. The second term is the Franck-Condon factor, that is discussed below in more detail. It leads to the well-known Franck-Condon progression of vibrational satellites that progress in the spectrum by the energy Vq of the normal mode under consideration. 

The nuclear frequency is related to the solvent and inner-shell reorganization energies as well as the corresponsing vibration frequencies. The electronic factor can be described on the basis of the Landau-Zener framework and is related to the electronic coupling matrix element... 

Here, = < /, V /y> is the electronic coupling matrix element and (FCWD) denotes the Franck-Condon weighted density of states. In the high-temperature regime — that is, when assuming that all vibrational modes are classical (fico, kgT ), the FCWD obeys a standard Arrhenius type of equation ... 

Values for many of the parameters in Heff cannot be determined from a spectrum, regardless of the quality or quantity of the spectroscopic data, because of correlation effects. When two parameters enter into the effective Hamiltonian with identical functional forms, only their sum can be determined empirically. Sometimes it is possible to calculate, either ab initio or semiempirically, the value of one second-order parameter, thereby permitting the other correlated parameter to be evaluated from the spectrum. Often, although the parameter definition specifies a summation over an infinite number of states, the largest part or the explicitly vibration-dependent part of the parameter may be evaluated from an empirically determined electronic matrix element times a sum over calculable vibrational matrix elements and energy denominators (Wicke, et al, 1972). 

Meg (in atomic units, IDebye = 0.3935 a.u.) is the electronic transition matrix element between the e and g electronic states, assuming the dipole length approximation, (ve is the energy normalized nuclear continuum wavefunction, and fj) is the initial state bound vibrational wavefunction. The overlap integral has units of cm1/2 (see Section 7.5). Note that 10 18 cm2=lMb... 

Chapters 2, 3, and 5 form the core of this book. Perturbations are defined and simple procedures for evaluating matrix elements of angular momentum operators are presented in Chapter 2. Chapter 3 deals with the troublesome terms in the molecular Hamiltonian that are responsible for perturbations. Particular attention is devoted to the reduction of matrix elements to separately evaluable rotational, vibrational, and electronic factors. Whenever possible the electronic factor is reduced to one- and two-electron orbital matrix elements. The magnitudes and physical interpretations of matrix elements are discussed in Chapter 5. In Chapter 4 the process of reducing spectra to molecular constants and the difficulty of relating empirical-parameters to terms in the exact molecular Hamiltonian are described. Transition intensities, especially quantum mechanical interference effects, are discussed in Chapter 6. Also included in Chapter 6 are examples of experiments that illustrate, sample, or utilize perturbation effects. The phenomena of predissociation and autoionization are forms of perturbation and are discussed in Chapters 7 and 8. 

Where the intermolecular potentiais are of the kind shown in Figure 6, the expression for the transition probability, again includes a vibrational matrix element of a similar form to that in equation (48). In the electronically adiabatic processes that were considered earlier, transition probabilities are extremely small, because enagy transfer requires a tunnelling process to carry the system between the parallel curves representing neighbouring vibronic states within the same electronic manifold. Now, however, curves cross and is replaced by a... 

In this equation, the Franck-Condon factors (the squared Franck-Condmi integrals) and the resonance condition [the delta function in Eq. (4)] have been absorbed into the spectral density 2>eet [135]. It can be factored into the line-shape functions for donor emission and acceptor absorption (this is only possible due to the assumption of local vibrational modes). In addition, the dipole approximation can be made for the electronic coupling matrix element ... 

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