External coulomb field

If the external field is the Coulomb field of a point charge located at the origin in particular, if... 

Within models of the sparkle family the effect of the external Coulomb field does not reduce to the renormalization of the orbital energies as it is within the RLMO model (see above). By contrast, the electron distribution also changes when the ligand molecules are put into the field. We model this by classical polarizability. Accordingly the difference between effective charge on atom A in the complex (polarized) and that in the free ligand (non-polarized) is ... 

We will describe, in some detail, one such modification, an effective Dirac equation (EDE) which was derived in a number of papers [7, 8, 9, 10]. This new equation is more convenient in many applications than the original BS equation, and we will derive some general formulae connected with this equation. The physical idea behind this approach is that in the case of a loosely bound system of two particles of different masses, the heavy particle spends almost all its life not far from its own mass shell. In such case some kind of Dirac equation for the light particle in an external Coulomb field should be an excellent starting point for the perturbation theory expansion. Then it is convenient to choose the free two-particle propagator in the form of the product of the heavy particle mass shell projector A and the free electron propagator... 

Fig. 1.6. Effective Dirac equation in the external Coulomb field...

EDE in the external Coulomb field in Fig. 1.6. The eigenfunctions of this equation may be found exactly in the form of the Dirac-Coulomb wave functions (see, e.g, [10]). For practical purposes it is often sufficient to approximate these exact wave functions by the product of the Schrodinger-Coulomb wave functions with the reduced mass and the free electron spinors which depend on the electron mass and not on the reduced mass. These functions are very convenient for calculation of the high order corrections, and while below we will often skip some steps in the derivation of one or another high order contribution from the EDE, we advise the reader to keep in mind that almost all calculations below are done with these unperturbed wave functions. 

Calculation of the nonlogarithmic polarization operator contribution is quite straightforward. One simply has to calculate two terms given by ordinary perturbation theory, one is the matrix element of the radiatively corrected external magnetic field, and another is the matrix element of the radiatively corrected external Coulomb field between wave functions corrected by the external magnetic field (see Fig. 9.13). The first calculation of the respective matrix elements was performed in [34]. Later a number of inaccuracies in [34] were uncovered [22, 23, 40, 43, 44, 45] and the correct result for the nonlogarithmic contribution of order a Za) EF to HFS is given by... 

The energy dissipation of a system containing free charges subjected to electric fields Is well known but this Indicates a non-equilibrium situation and as a result a thermodyanmlc description of the FDE Is Impossible. Within the framework of interionic attraction theory Onsager was able to derive the effect of an electric field on the Ionic dissociation from the transport properties of the Ions In the combined coulomb and external fields (2). It is not improper to mention here the notorious mathematical difficulty of Onsager s paper on the second Wien effect. 

Figure 1. Orbital period of an electron moving in a Coulomb field, the time scales of some internal and external perturbations [3a-3d], and the observed (shorter, see below) lifetime for the polyatomic molecule known as BBC [4]. Note that at the highermost values of n the decay lifetime begins to shorten cf. Fig. 4.

The MQDT picture [39, 40] of a Rydberg molecule in an electric field [41, 42] reveals much about the esential physics of the Rydberg stabilization problem. The space of the electron is divided primarily into three regions. The effects of the external field are only felt in the outermost region IB, where the total potential for the electron is a sum of Coulombic and external field terms (in a.u.)... 

The PF formalism treats the electron-hole pair dissociation as a one step carrier escapes from the Coulomb field of the countercharge due to external field-assisted thermal activation over the barrier. The two steps in the formation of a free carrier pair, as shown in Fig. 48, are undistinguishable. In other models of the exciton dissociation process, these two steps are separable, and the overall quantum efficiency expressed by the product... 

The electronic polarisability of a spherical atom may be calculated in a number of simplified ways. In the oldest approximation, an atom is regarded as a conductive sphere of radius R, when the polarisability may be shown to be 4k 0R3, a quantity that is closely related to the actual volume of a molecule. In the more realistic semi-classical Bohr model of a hydrogen atom, the application of a field normal to the plane of the electron orbit, radius R, will produce a small shift, — x, in the orbit, as shown in Fig. 2.2. To a first approximation the distance of the orbit from the nucleus will still be R and the dipole moment p. induced in the atom will have magnitude ex. At equilibrium, the external field acting on the electron is balanced by the component of the Coulombic field from the positive nucleus in the field direction ... 

In the absence of a driving force (e.g., an externally applied electric field), no direction in space from the central ion is privileged. The Coulombic field of the central ion has spherical symmetry and therefore the probability of finding, say, a negative ion at a distance r from the reference ion is the same irrespective of the direction in which the point r lies. On this basis, it was shown that the ionic cloud was spherically symmetrical (see Section 3.8.2). 

Owing to rapid experimental progress in the field of laser physics, ultra-short laser pulses of very high intensity have become available in recent years. The electric field produced in such pulses can reach or even exceed the strength of the static nuclear Coulomb field. If an atomic system is placed in the focus of such a laser pulse one observes a wealth of new phenomena [240] which cannot be explained by perturbation theory. In this case a non-perturbative treatment, i.e., the solution of the full TDKS equations (39)-(41) is mandatory. The total external potential seen by the electrons is given by... 

According to Equation (1.15) the exact first-order vacuum polarization potential induced by the static external Coulomb field of a nucleus reads (in the Feynman... 

The significance of the two other quantum numbers only becomes evident if the degeneracy is removed by some perturbation (due to deviations from the Coulomb field, introduction of the relativistic variation of mass, presence of an external field, or some other cause). We can, however, gain an idea of the meaning of the quantum numbers from the purely geometrical point of view by considering the elliptic orbit. If, as in 1 (p. 99), we denote the radius of the first circular Bohr orbit for Z = 1 by... 

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