The Real Electron

In this form there is a close correspondence with the equations of hydrodynamics, or vortex-free flow of a fluid under the influence of conservative forces. Equation (12) resembles a hydrodynamic continuity equation if a2 is considered to be a density and if the stream velocity v = V(j). As V x v = 0, 

The first term on the right is interpreted as a force density and the second corresponds to a quantity like — f with pressure p and mass density p, or a function of an internal stress gradient rather than actual stress, as in hydrodynamics. 

In a refined form of the theory [37] the same quantity features as the well known quantum potential. The notion of a particle emerges in this theory in the form of a highly localized inhomogeneity that moves with the local fluid velocity, v(x, t), thus as a stable dynamic structure that exists in the fluid, for example, as a small stable vortex or a pulse-like distortion. To explain why the causal theory needs probability densities it is argued [37] that the Madelung fluid must experience more or less random fluctuations in its motion to account for irregular turbulence. The turbulence necessitates a wave theory to describe the motion of vortices embedded in the fluid. The particle velocity is therefore not exactly VS/m, nor is the density exactly 

After separating real and imaginary parts in the usual way, two equations are obtained  

The first is a relativistic Hamilton-Jacobi (HJ) equation for a particle with variable rest mass [101] 

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In the vector potential approach [6], the (real) electronic wave function (4>) is multiplied by a complex phase factor/(4>), defined such that... 

Within the approximation of the effective mass, consideration of the field created by the condensed media is confined to substitution of the real electron mass by the effective mass. Precise calculation of the effective mass is equivalent to solution of the Schrodinger equation with the consideration of the field created by the medium, and, consequently, as noted before, is hardly possible. Thus, as far as the problem of electron tunneling is concerned, the effective mass must be considered as a phenomenological parameter. In the case of tunneling with the energy I of the order of 1-5 eV, the field created by the medium apparently increases considerably the probability of electron tunneling, and the effective mass of electron can be noticeably lower than the real mass. 

To utilize Eq. 7.11, Kohn and Sham introduced the idea of a fictitious reference system of noninteracting electrons which give exactly the same electron density distribution as the real system has. Addressing electronic kinetic energy, let us define the quantity A(T[p ] ) (don t confuse Greek delta A, an increment, with the differential operator del V) as the deviation of the real electronic kinetic energy from that of the reference system ... 

Apart from the ill-definitiveness of the inverse vibrational problem two principal objections are usually posed against the standard VFF model [5], First, the neglect of the long range interactions is not always physically justified and often contradicts with the real electronic structure of the molecule under study. Second, the transferability of force-constants is still a disputable topic, especially when the force-constants are transferred between neutral molecules and the corresponding ionized forms, or between conformational isomers [6, 7],... 

It should be understood that the choice of this name, based mainly on practical nomenclature considerations, does not define the real electronic structure of the molecule, which will be discussed later in Section VI. 

All in all, the bi-orthogonal dilated electron propagator offers a simple extension of the real electron propagator technique and with the incorporation of higher order decouplings like the E3, E ADC(3) etc. and suitably large and flexible basis sets should offer same power and effectiveness in the treatment of metastable anions and cations as done by its real counterpart for stable bound systems. 

Qualitatively new results concerning the determination of the neutrino rest mass were obtained in 1980, when the data reduction of a set of experiments performed for more than 5 years at the Moscow Institute of Theoretical and Experimental Physics (ITEP) (Lubimov et al, 1980, 1981) for the first time give the lower limit on the neutrino rest mass. The /J-electron source was a doubly tritrated amino acid, valine (C5HnN02). The early reductions of the results obtained in these experiments have shown that the confidence interval for the neutrino mass substantially depends on the chosen theoretical model. Since the real electron excitation spectrum of the / source was not known, the experimental data were reduced for two model cases ... 

The real key, though, is the definition of the first term of Eq. (1.47). Kohn and Sham defined it as the kinetic energy of noninteracting electrons whose density is the same as the density of the real electrons, the true interacting electrons. The last term is called the exchange-correlation functional, and is a catchaU term to account for all other aspects of the true system. 

In principle, the Kohn-Sham orbitals obtained as solutions to Eq. (3.39) are for noninteracting particles, not the real electrons, but in practice this difference is often ignored. 

In reality, the electronic states of Re(I) complexes are mixed with each other by configuration interactions, and the real electronic states are best viewed as composites of the virtual pure electronic states such as MLCT and rni excited states, etc. In this chapter, the real electronic states will be represented by the virtual pure state which has the largest contribution to the excited state properties. 

The U(R2, C ) is the Born-Oppenheimer energy, which now explicitly depends on the c coordinates and consists of the nucleus-nucleus, electron-nucleus, and electron-electron interaction energy terms. Note that the real electron kinetic energy is included in this last term. The equation of motion for the particles (both real and fictitious) are then obtained from the extended Lagrangian (Eq. [71]) and reads as follows ... 

The close similarity of CTC of naphthalene adsorbed on the decationated zeolite, porous silica and alumina allows us to make an assumption that the acid sites of Bronsted and Lewis type related to the coordinatively unsaturated silicon and aluminium ions are the real electron-accepting sites. 

The Hartree-Fock method generates solutions to the Sdirddinger elation where the real electron-electron interaction is replaced by an average interaction (Chapter 3). In a sufficiently large basis, the HP wave function is able to account for 99% of the total... 

Mead and Truhlar [52] introduced an elegant way of incorporating the geometric phase effect, namely the vector potential approach. In this method, the real electronic wave function 4>(a), where a is any internal angular coordinate describing the motion around the Cl, is multiplied by a complex phase factor c(a) to ensure the single-valuedness of the new complex electronic wave function ... 

Many molecules can be represented by two or more Lewis structures that differ only in the placement of electrons. In such cases the electrons are delocalized, and the real electron distribution is a composite of the contributing Lewis structures, each of which is called a resonance form. The rules for resonance are summarized in Table 1.5. 

Although the reduced forms of cytochromes c-552(m) (22.3 kDa) (or cytochrome c4) and c-550(m) (51 kDa) are also oxidized with molecular oxygen by the catalysis of cytochrome c oxidase, it has not yet been verified whether these cytochromes are reduced with ferrous ion by the catalysis of Fe(II)-cytochrome c oxidoreductase. On the basis of the studies of DNA that encodes the redox proteins of A. ferrooxidans, it is suggested that cytochrome c4 is the direct electron donor for cytochrome c oxidase in vivo (Appia-Ayme et al., 1999). However, as already mentioned, the reactivity with cytochrome c4 of cytochrome c oxidase is much lower than that of cytochrome c-552(s) (14 kDa) and is depressed with sulfate, while the enzymatic reaction of cytochrome c-552(s) (14 kDa) is stimulated by the salt. So cytochrome c-552(s) (14 kDa) seems to function as the real electron donor for the oxidase, if cytochrome c oxidase catalyzes the reduction of molecular oxygen at the outside of the plasma membrane as already mentioned (see also Fig. 5.2). 

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