Time-dependent perturbation theory, electron

Time-dependent perturbation theory, electron nuclear dynamics (END), molecular systems, 340-342... 

As for electron transfer in the normal region, based on the results of time dependent perturbation theory, electron transfer in the inverted or excited state decay region is also determined by the... 

The interaction of a molecular species with electromagnetic fields can cause transitions to occur among the available molecular energy levels (electronic, vibrational, rotational, and nuclear spin). Collisions among molecular species likewise can cause transitions to occur. Time-dependent perturbation theory and the methods of molecular dynamics can be employed to treat such transitions. 

The tools of time-dependent perturbation theory can be applied to transitions among electronic, vibrational, and rotational states of molecules. 

Time dependent perturbation theory provides an expression for the radiative lifetime of an excited electronic state, given by Tr ... 

The presence of the electron acceptor site adjacent to the donor site creates an electronic perturbation. Application of time dependent perturbation theory to the system in Figure 1 gives a general result for the transition rate between the states D,A and D+,A. The rate constant is the product of three terms 1) 27rv2/fi where V is the electronic resonance energy arising from the perturbation. 2) The vibrational overlap term. 3) The density of states in the product vibrational energy manifold. 

In Sect. 2.1, the electron transfer rate was defined as the Boltzmann average of transition probabilities, which were calculated through time-dependent perturbation theory by using the Born-Oppenheimer and Frank-Condon approx-... 

Some authors have described the time evolution of the system by more general methods than time-dependent perturbation theory. For example, War-shel and co-workers have attempted to calculate the evolution of the function /(r, Q, t) defined by Eq. (3) by a semi-classical method [44, 96] the probability for the system to occupy state v]/, is obtained by considering the fluctuations of the energy gap between and 11, which are induced by the trajectories of all the atoms of the system. These trajectories are generated through molecular dynamics models based on classical equations of motion. This method was in particular applied to simulate the kinetics of the primary electron transfer process in the bacterial reaction center [97]. Mikkelsen and Ratner have recently proposed a very different approach to the electron transfer problem, in which the time evolution of the system is described by a time-dependent statistical density operator [98, 99]. 

Bardeen considers two separate subsystems first. The electronic states of the separated subsystems are obtained by solving the stationary Schrodinger equations. For many practical systems, those solutions are known. The rate of transferring an electron from one electrode to another is calculated using time-dependent perturbation theory. As a result, Bardeen showed that the amplitude of electron transfer, or the tunneling matrix element M, is determined by the overlap of the surface wavefunctions of the two subsystems at a separation surface (the choice of the separation surface does not affect the results appreciably). In other words, Bardeen showed that the tunneling matrix element M is determined by a surface integral on a separation surface between the two electrodes, z = zo. 

The Time Dependent Processes Section uses time-dependent perturbation theory, combined with the classical electric and magnetic fields that arise due to the interaction of photons with the nuclei and electrons of a molecule, to derive expressions for the rates of transitions among atomic or molecular electronic, vibrational, and rotational states induced by photon absorption or emission. Sources of line broadening and time correlation function treatments of absorption lineshapes are briefly introduced. Finally, transitions induced by collisions rather than by electromagnetic fields are briefly treated to provide an introduction to the subject of theoretical chemical dynamics. 

The simplest approach to understanding the radiation- (light-) induced transition between electronic states is to invoke time-dependent perturbation theory. Thus, one starts from the time-dependent Schrodinger equation... 

The CT excitation and the ensuing CR are described within the framework of the time-dependent perturbation theory in the electronic coupling V. The time dependent CR rate constant is [3] ... 

The discussion in the previous section was helpful in identifying the factors at the molecular level which are involved when electron transfer occurs. Two different theoretical approaches have been developed which incorporate these features and attempt to account for electron transfer rate constants quantitatively. The first, by Marcus34 and Hush,35 is classical in nature, and the second is based on quantum mechanics and time dependent perturbation theory. The theoretical aspects of electron transfer in chemical36-38 and biological systems39 have been discussed in a series of reviews. 

Returning to equation (38), in the limit that ve vn, Ke = 1 and vet = vn. Electron transfer reactions that fall into this domain where the probability of electron transfer is unity in the intersection region have been called adiabatic by Marcus. Reactions for which Kei < 1, have been called non-adiabatic . In the limit that ve 2vn and e = vjvn, the pre-exponential term for electron transfer is given by vet = ve. This was the limit assumed in the quantum mechanical treatment using time dependent perturbation theory. 

For a quantitative treatment of establishing connections between vibronic coupling and vibrational progressions in electronic spectra, band profiles from vibronic wavefunctions must be calculated using common procedures of time-dependent perturbation theory and Fermi s golden rule [57], For emission, e.g., the transition rate which is the transition probability per unit time summed over... 

Resonances are common and unique features of elastic and inelastic collisions, photodissociation, unimolecular decay, autoionization problems, and related topics. Their general behavior and formal description are rather universal and identical for nuclear, electronic, atomic, or molecular scattering. Truhlar (1984) contains many examples of resonances in various fields of atomic and molecular physics. Resonances are particularly interesting if more than one degree of freedom is involved they reflect the quasi-bound states of the Hamiltonian and reveal a great deal of information about the multi-dimensional PES, the internal energy transfer, and the decay mechanism. A quantitative analysis based on time-dependent perturbation theory follows in the next section. 

Near the TS things change. The rapid evolution of the light components of the system (electrons and H atoms involved in a transfer process) makes the adiabatic approximation questionable. Also the sudden time dependent perturbation we introduced in Section 1.1.3 to describe solvent effects on electronic transitions is not suitable. We are considering here an intermediate case for which the time dependent perturbation theory does not provide simple formulae to support our intuitive considerations. Other descriptions have to be defined. 

In quantum mechanics the definition of molecular polarizabilities is given through time-dependent perturbation theory in the electric dipole approximation. These expressions are usually given in terms of sums of transition matrix elements over energy denominators involving the full electronic structure of the molecule [42]. 

For an elaborated analysis of the relations between structure and hyperpolarizabilities, one has to start from the electronic wavefunctions of a molecule. By using time-dependent perturbation theory, sum-over-states expressions can be derived for the first and second-order hyperpolarizabilities j3 and y. For / , a two-level model that includes the ground and one excited state has proven to be sufficient. For y the situation is more complicated. 

Three landmark papers on the application of time-dependent perturbation theory to electrochemical problems were published in rapid succession by - Levich and - Dogonadze in 1959 [iii], - Gerischer in 1960 [iv], and McConnell in 1961 [v]. A very large literature has subsequently sprung from these works, driven by developments in scanning tunneling microscopy, molecular electronics, and biological electron transfer. 

The electron hopping frequency may be estimated from time-dependent perturbation theory. If Hab is treated as a constant perturbation, the system will start to oscillate between the two diabatic states once the perturbation is turned on. In a bimolecular reaction, for example, the perturbation is turned on upon formation of the precursor complex, while in a covalently attached (bridged) binuclear system it can be turned on upon reduction (oxidation) of one end of the fully oxidized (reduced) system by an external reagent or by photoexcitation. If the system is in the diabatic reactant state at / = 0, then the probability of it being in the product state at some later time t is given by the Rabi formula [27]. 

The time-dependent perturbation theory of the rates of radiative ET is based on the Born-Oppenheimer approximation [59] and the Franck Condon principle (i.e. on the separation of electronic and nuclear motions). The theory predicts that the ET rate constant, k i, is given by a golden rule -type equation, i.e., it is proportional to the product of the square of the donor-acceptor electronic coupling (V) and a Franck Condon weighted density of states FC) ... 

The external fields induce forced oscillations in the electron cloud. The interaction is described in terms of the time-dependent Hamiltonian within the framework of propagator methods [3] or, equivalently, introducing time-dependent perturbation theory [22, 23]. Relaxing condition (2) for the free wave, the general form of the Hamiltonian becomes, neglecting electron spin. 

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