Coulomb retarded

Diflfiision-controlled reactions between ions in solution are strongly influenced by the Coulomb interaction accelerating or retarding ion diffiision. In this case, die dififiision equation for p concerning motion of one reactant about the other stationary reactant, the Debye-Smoluchowski equation. 

The driving force of the reaction between a radical and an aromatic cation-radical is based on their mutual affinity because both species possess unpaired electrons. There is no coulombic attraction and no coulombic repulsion in this case. The reaction between an anion and an aromatic cation-radical involves electrostatic attraction as the driving force. It necessarily proceeds through a contact (intimate) ion pair Ar+, A. It is obvious that disintegration of the mentioned ion pair has to retard or even prevent ArA formation. 

Hpg c represents the relativistic correction of Coulomb interaction due to the retardation... 

For a quasi-relativistic framework as relevant to chemistry (21), we may neglect the magnetic retardation between the electrons and the nuclei and therefore employ standard Coulombic interaction operators for the electrostatic interaction. The interaction between the electrons and the nuclei is not specified explicitly but we only describe the interactions by some external 4-potential. For the sake of brevity this 4-potential shall comprise all external contributions. Explicit expressions for the interaction between electrons and nuclei will be introduced at a later stage. Furthermore, we can neglect the relativistic nature of the kinetic energy of the nuclei and employ the non-relativistic kinetic energy operator denoted as hnuc(I),... 

Term I denotes the classical electron-electron interaction which is related to Darwin s classical expression when expressed in terms of momenta rather than velocities (21). It comprises Coulombic interactions, magnetic, and retardation terms. Term... 

When an ion with charge q (coulombs) is placed in an electric field E (V/m), the force on the ion is qE (newtons). In solution, the retarding frictional force is /uep, where ncp is the velocity of the ion and/is the friction coefficient. The subscript ep stands for electrophoresis. The ion quickly reaches a steady speed when the accelerating force equals the frictional force ... 

For numerical evaluation (to sum over the entire spectrum of Dirac equation) B-splines are used [28], in particular the version developed by I.A. Goidenko [29]. Earlier the full QED calculations were carried out only for the ground (lsi/2)2 state He-like ions for the various nuclear charges Z. At that ones used either B-splines or the technique of discretization of radial Dirac equations [27]. As well as in [27] we used the Coulomb gauge. For control we reproduced the results of the calculation of (lsi/2)2 state and compared them with ones of [27]. Coulomb-Coulomb interaction is reproduced for every Z with the accuracy, on average, 0.01 %, Coulomb-Breit is with the accuracy 0.05 % and Breit-Breit (with disregarding retardation) is with the accuracy 0.1%. The small discrepancy is explained by the difference in the numerical procedures applied in [27] and in this work. 

It is easy to recognize in < >(r) the tensorial expression for the retarded field created at the point r by a dipole located at the origin r = 0. The instantaneous term in 1 jr3 coincides, in the approximation of the retarded effects, with the electrostatic field and reestablishes the coulombic dipole-dipole interaction. We shall investigate < >(r) at distances short compared to A, which allows us to expand (1.37) in powers of cor/c ... 

This approach clearly distinguishes two ranges of interaction At the scale r < A, where the electrostatic interaction dominates, and at the scale r > /, where the retardation effects dominate. This scale property justifies the separation, implicit in the Coulomb gauge, between instantaneous terms and retarded terms. However, the electric-dipole gauge shows that these two distinct aspects of the electromagnetic interaction are physically undissoci-able, even though it is possible in many problems to omit retardation effects. 

The expression (1.69) for e is quite general in the sense that it gives the response of the crystal to an external field of any wave vector. In particular the poles of s(K, (o) provide, over the whole Brillouin zone, the dispersion curves of the new elementary excitations built up by coulombic and retarded interactions. 

Umklapp process In the interaction of a continuous wave (photon, electron, etc.) with the lattice, the quasi-momentum of the wave is conserved, modulo a vector in the reciprocal lattice. The introduction of these quanta of momentum leads to the Umklapp process. In many macroscopic treatments the matter is treated as a continuous medium and Umklapp processes are neglected. In our treatment, Umklapp processes are included in the coulombic interactions (calculation of the local field), but implicitly omitted in the retarded interactions, since we dropped the term (cua/c)2 in (1.64). 

Outside of a small region around the center of the Brillouin zone, (the optical region), the retarded interactions are very small. Thus the concept of coulombic exciton may be used, as well the important notions of mixure of molecular states by the crystal field and of Davydov splitting when the unit cell contains many dipoles. On the basis of coulombic excitons, we studied retarded effects in the optical region K 0, introducing the polariton, the mixed exciton-photon quasi-particle, and the transverse dielectric tensor. This allows a quantitative study of the polariton from the properties of the coulombic exciton. 

In what follows, we present in Section IV.A a theory of the effects of weak disorder on the retarded interactions of 2D strong dipolar excitons, and in Section IV.B we analyze the effects of stronger disorders on the coulombic interactions, calculating the density of states and absorption spectra in 2D lattices, in the framework of various approximations of the mean-field theory. 

Assemblies of molecules can be seen in a first approximation as assemblies of transition dipoles subject to electromagnetic interactions. If only the coulombic part of the interaction is kept, new eigenstates of the assembly are found with real energies (the excitonic states). To account for the spontaneous emission, we have to include the retardation (to consider the exciton-photon coupling), which amounts to taking the classical problem of... 

To conclude, we can draw an analogy between our transition and Anderson s transition to localization the role of extended states is played here by our coherent radiant states. A major difference of our model is that we have long-range interactions (retarded interactions), which make a mean-field theory well suited for the study of coherent radiant states, while for short-range 2D Coulombic interactions mean-field theory has many drawbacks, as will be discussed in Section IV.B. Another point concerns the geometry of our model. The very same analysis applies to ID systems however, the radiative width (A/a)y0 of a ID lattice is too small to be observed in practical experiments. In a 3D lattice no emission can take place, since the photon is always reabsorbed. The 3D polariton picture has then to be used to calculate the dielectric permittivity of the disordered crystal see Section IV.B. 

In the intermediate domain of values for the parameters, an exact solution requires the specific inspection of each configuration of the system. It is obvious that such an exact theoretical analysis is impossible, and that it is necessary to dispose of credible procedures for numerical simulation as probes to test the validity of the various inevitable approximations. We summarize, in Section IV.B.l below, the mean-field theories currently used for random binary alloys, and we establish the formalism for them in order to discuss better approximations to the experimental observations. In Section IV.B.2, we apply these theories to the physical systems of our interest 2D excitons in layered crystals, with examples of triplet excitons in the well-known binary system of an isotopically mixed crystal of naphthalene, currently denoted as Nds-Nha. After discussing the drawbacks of treating short-range coulombic excitons in the mean-field scheme at all concentrations (in contrast with the retarded interactions discussed in Section IV.A, which are perfectly adapted to the mean-field treatment), we propose a theory for treating all concentrations, in the scheme of the molecular CPA (MCPA) method using a cell... 

The problem is to discuss the generalized polarizability ae(1.49) with the matrix a 1 not commuting with that of the dipolar interactions, 0. To show that the pure retarded interactions may be discarded in the dynamics of mixed crystals, we assume here that the coulombic interactions are suppressed in (ft. The interaction tensor is then reduced to its retarded term (1.74). Then the dispersion is given by (1.35) ... 

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