Time-dependent perturbation theory, electron systems

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

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. 

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. 

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. 

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 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 Marcus and Hush, is classical in nature, and the second is based on quantum mechanics and time dependent perturbation theory. The theoretical aspects of electron transfer in chemicaP and biological systems have been discussed in a series of reviews. 

Taking advantage of the fact that the electronic states within a band are only weakly coupled, the non-adiabatic couplings are often treated using time-dependent perturbation theory to first-order.This allows to characterize the dynamical evolution of the system implicitly, within a time-independent framework. The typical derivation for the vibrational... 

A particularly convenient improved approximation to this end can be obtained by use of self-consistent, first order, time-dependent perturbation theory. The essential physics to be included is that the external field distorts the atomic charge cloud (by admixture of excited orbitals) which in turn creates an electrostatic potential acting on the system, The self-consistent response of the electrons produces a mean field which reflects the atomic dielectric properties and alters the photoionization amplitudes. If this linear response approach is applied to the HFA one obtains precisely the RPAE, In what follows we consider the same approximation applied to the LDA. Given this parallelism, emphasis will be placed on direct comparisons with the RPAE,... 

The reaction of a many-electron system to a time-dependent perturbation is almost as important as its adjustment to a static time-independent environment change. The theory of the first-order response of a system to an oscillating field follows very similar lines to the perturbation theory of the last chapter. 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

The study of behavior of many-electron systems such as atoms, molecules, and solids under the action of time-dependent (TD) external fields, which includes interaction with radiation, has been an important area of research. In the linear response regime, where one considers the external held to cause a small perturbation to the initial ground state of the system, one can obtain many important physical quantities such as polarizabilities, dielectric functions, excitation energies, photoabsorption spectra, van der Waals coefficients, etc. In many situations, for example, in the case of interaction of many-electron systems with strong laser held, however, it is necessary to go beyond linear response for investigation of the properties. Since a full theoretical description based on accurate solution of TD Schrodinger equation is not yet within the reach of computational capabilities, new methods which can efficiently handle the TD many-electron correlations need to be explored, and time-dependent density functional theory (TDDFT) is one such valuable approach. 

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