Valence charge density distribution

The evidence for covalent T-Al bonding can also be visualized in real space by the valence charge density distribution as an example, this is shown for YbNiAl with hexagonal ZrNiAl structure in fig. 6. The accumulation of charge is clearly discernible between A1 and Ni with the charge density shifted to the more electronegative nickel atoms. 

Figure 15. Radial charge density plot for the resonant p-type virtual orbital for dilation angles 9 = 0.0 and 6 = 90pt (0.42 radians) in e-Be scattering. The role of optimal theta in the accumulation of electron density near the nucleus is clearly seen. In the inset, the maximum is seen to occur at rmaz — 2.5 a.u., very close to that for the rmax of the outer valence 2s orbital, seen in fig. 14- Though a cursory look at the nodal pattern identifies this as a 4P orbital, the dominant contribution to the charge density distribution is mainly of 2p-iype.

None of these implementations can be used to study effects due to variar tion of the finite nucleus model, due to their limitation to a single finite nucleus model. Of course, it is unlikely that such variations lead to significant changes in the chemical behaviour of atoms and molecules, e.g., reaction enthalpies, valence electronic charge density distribution etc. However, the finer details of the electron distribution in the vicinity of heavy atomic nuclei will be more sensitive to the variation of the finite nucleus model, but this is clearly a field in the area of atomic and nuclear physics. 

It means that for electrons which satisfy condition of extreme nonadiabaticity (antiadiabaticity with respect to interacting phonon mode r in particular direction of reciprocal lattice where the gap in one-electron spectrum has been opened), the electron (nonadiabatic polaron)-renormalized phonon interaction energy equals zero. Expressed explicitly, in the presence of external electric potential, dissipation-less motion of relevant valence band electrons (holes) on the lattice scale can be induced at the Fermi level (electric resistance p = 0). At the same time, the motion of nuclei remains bound to circumferential revolution over distorted, energetically equivalent, configurations. The electrons move in a form of itinerant-mobile bipolarons, i.e. as a polarized cloud of inter-site charge density distribution- sequence b, d, e, f, b, d, e, f. in Fig. 27.6. For temperature increase, thermal excitations of valence band electrons to conduction band induce sudden transition from the antiadiabatic state to adiabatic state at T — 7, i.e. < AEd Rd) holds and the system is... 

Figure Al.3.22. Spatial distributions or charge densities for carbon and silicon crystals in the diamond structure. The density is only for the valence electrons the core electrons are omitted. This charge density is from an ab initio pseudopotential calculation [27].

We have described in this paper the first implementation of this Bayesian approach to charge density studies, making joint use of structural models for the atomic cores substructure, and MaxEnt distributions of scatterers for the valence part. Used in this way, the MaxEnt method is safe and can usefully complement the traditional modelling based on finite multipolar expansions. This supports our initial proposal that accurate charge density studies should be viewed as the late stages of the structure determination process. 

A simple modification of the IAM model, referred to as the K-formalism, makes it possible to allow for charge transfer between atoms. By separating the scattering of the valence electrons from that of the inner shells, it becomes possible to adjust the population and radial dependence of the valence shell. In practice, two charge-density variables, P , the valence shell population parameter, and k, a parameter which allows expansion and contraction of the valence shell, are added to the conventional parameters of structure analysis (Coppens et al. 1979). For consistency, Pv and k must be introduced simultaneously, as a change in the number of electrons affects the electron-electron repulsions, and therefore the radial dependence of the electron distribution (Coulson 1961). 

Aromatic substitution reactions are often complicated and multistep processes. A correlation, however, in many cases can be found between the charged attacking species and the electron density distribution in the molecule attacked during electrophilic and nucleoph c substitution. No such correlation is expected in radical substitution where the attacking particles are neutral, rather a correlation between the reactivities of separate bonds and a free valency index of the bond order. This allows the prediction of the most reactive bonds. Such an approach has been used by researchers who applied quantum calculations to estimate the reactivities of the isomeric thienothiophenes and to compare them with thiophene or naphthalene. " Until recently quantum methods for studying reactivities of aromatics and heteroaromatics were developed mainly in the r-electron approximation (see, for example, Streitwieser and Zahradnik ). The M orbitals of a sulfur atom were shown not to contribute substantially to calculations of dipole moments, polarographic reduction potentials, spin-density distribution, ... 

Several formulations were proposed [65, 66], but the intuitive notation introduced by Hansen and Coppens [67] afterwards became the most popular. Within this method, the electron density of a crystal is expanded in atomic contributions. The expansion is better understood in terms of rigid pseudoatoms, i.e., atoms that behave stmcturally according to their electron charge distribution and rigidly follow the nuclear motion. A pseudoatom density is expanded according to its electronic stiucture, for simplicity reduced to the core and the valence electron densities (but in principle each atomic shell could be independently refined). Thus,... 

Two approaches have been applied to estimate the net charges of atoms from the observed electron-density distribution in crystals. The first method is a direct integration of observed density in an appropriate region around an atom (hereafter abbreviated as DI method) (64). The second is the so-called extended L-shell method (ELS method) (19, 81) in which a valence electron population of an atom is calculated by a least-squares method on the observed and calculated structure amplitudes. 

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