States one-electron

Fledin L and Lundqvist S 1969 Effects of electron-electron and electron-phonon interactions on the one-electron states of solids Solid State Phys. 23 1... 

Knowledge of the underlying nuclear dynamics is essential for the classification and description of photochemical processes. For the study of complicated systems, molecular dynamics (MD) simulations are an essential tool, providing information on the channels open for decay or relaxation, the relative populations of these channels, and the timescales of system evolution. Simulations are particularly important in cases where the Bom-Oppenheimer (BO) approximation breaks down, and a system is able to evolve non-adiabatically, that is, in more than one electronic state. 

The expression for the force on the nuclei, Eq. (89), has the same form as the BO force Eq. (16), but the wave function here is the time-dependent one. As can be shown by perturbation theory, in the limit that the nuclei move very slowly compared to the electrons, and if only one electronic state is involved, the two expressions for the wave function become equivalent. This can be shown by comparing the time-independent equation for the eigenfunction of H i at time t... 

If more than one electronic state is involved, then the electronic wave function is free to contain components from all states. For example, for non-adiabatic systems the elecbonic wave function can be expanded in the adiabatic basis set at the nuclear geometry R t)... 

Conical intersections, introduced over 60 years ago as possible efficient funnels connecting different elecbonically excited states [1], are now generally believed to be involved in many photochemical reactions. Direct laboratory observation of these subsurfaces on the potential surfaces of polyatomic molecules is difficult, since they are not stationary points . The system is expected to pass through them veiy rapidly, as the transition from one electronic state to another at the conical intersection is very rapid. Their presence is sunnised from the following data [2-5] ... 

As is the case for diatomic molecules, rotational fine structure of electronic spectra of polyatomic molecules is very similar, in principle, to that of their infrared vibrational spectra. For linear, symmetric rotor, spherical rotor and asymmetric rotor molecules the selection mles are the same as those discussed in Sections 6.2.4.1 to 6.2.4.4. The major difference, in practice, is that, as for diatomics, there is likely to be a much larger change of geometry, and therefore of rotational constants, from one electronic state to another than from one vibrational state to another. 

Equations (11-477) and (11-476) impose restrictions on the form of jft(x) and / (x). As a consequence of these equations, we also have the result that if p, > is a one-electron state... 

However, we have no guarantee that the current operator defined by Eq. (11-490) will leave Eq. (11-483) satisfied. Stated differently, we have no guarantee that the charge -e, which occurs in the definition (11-490) in the current operator, is the same as the total charge of the one-electron state. In order to be able to satisfy this requirement we shall add another counter term of the form L"Z2A(x) to the expression (11-490) of the current operator. The addition of such a term does not affect Eq. (11-491). Hence, finally, the current operator is defined as... 

In other words, there are no observable consequences of the one-particle systems in the absence of external perturbations besides these the mass of the one-electron state is m, that of the photon is zero, the electric charge of the negaton is — e,10 and that of the positon, +e. 

Our results fit also with a previous investigation (9) on polyenes based on a version of the 2h-lp Cl scheme restricted to the virtual one-electron states generated by a minimal basis. In our case, however, the fragmentation of lines into satellites is much more pronounced. The reason lies in the size-consistency of the ADC[3] approach (as contrasted with the size-inconsistency of any truncated form of Cl (27d), in the full handling of the virtual space, and (10) in the inclusion of correlation corrections to the reference ground state, leading to (37) a net reduction of the quasi-particle band gap of conjugated polymers. 

Reactions occurring on two (or more) electronic states can lead to the same product asymptote. These pathways may occur if more than one electronic state correlates adiabatically to the same asymptote (e.g., single or triplet coupling of two approaching species), or if nonadiabatic transition(s) move population from one state to another. Here, I make the distinction that products of the same structural formula do not represent the same exit channel if they are produced in different electronic states. For example, in the reaction... 

For the description of the 3sel continua and of the higher 3pnl resonances, the bases include a large number of configurations of the forms 35Xn/)5 PXni), where 3.s,3p are the lowest one-electron states in the field of the SCF core. 

In view of very small spacing of the energy levels, one may not restrict oneself by consideration of the electron transfer from only one electron state in the metal, and all energy spectra shonld be taken into account. However, the entire process is composed of transitions with the participation of individnal energy levels 8. Therefore, the electron transfer from an energy level s to the reactant located a distance X from the electrode snrface in the solntion is considered first (Fig. 34.3). 

Of course the shape of the potential energy surface is a reflection of the electronic structure, which differs between one electronic state and another. We now introduce... 

Hedin, L. and Lundqvist, S. (1969) Effects of electron-electron and electron-phononinteraction on the one-electron states of solids, In Solid state Physics, Eherenreich, H., Seitz, F. and Turnbull, D. (Eds.), Academic, New York,Vol. 23, pp. 1-181. 

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