Central Coulomb field

The Schrodinger wave equation that describes the motion of an electron in an isolated hydrogen atom is a second-order linear differential equation that may be solved after specification of suitable boundary conditions, based on physical considerations. The solution to the equation, known as a wave function provides an exhaustive description of the dynamic variables associated with electronic motion in the central Coulomb field of the proton. 

The linear combination of atomic orbitals (LCAO) used as an interpolation method to derive the symmetry properties of energy bands in crystals [9] has as coefficients the values of plane waves, exp(ik.R), at the positions R of atoms in the unit cell. Rather than a LCAO the derived wave function therefore is a linear combination of Bloch sums with the full space-group symmetry of the crystal. Atomic orbitals, by contrast, are modes in a spherically symmetrical central Coulomb field. 

The atomic shell phenomena (electrons in a central Coulomb field) can be successfully treated with quantum-mechanical methods. In the case of atomic nuclei, the situation is more complicated, because here there is no well-defined central field, the average field of nucleons substantially differs from the well-known Coulomb field (the nuclear forces are very complicated), and there are two different types of nucleons in nuclei (p and n). 

In an attempt to overcome the above-mentioned difficulty, Dirac discovered an equation now well known as the Dirac equation which is essentially a relativistic analog of the Schrodinger equation. The resulting equation for a single electron in a central Coulombic field is... 

Thus, the state of each electron in a many-electron atom is conditioned by the Coulomb field of the nucleus and the screening field of the charges of the other electrons. The latter field depends essentially on the states of these electrons, therefore the problem of finding the form of this central field must be coordinated with the determination of the wave functions of these electrons. The most efficient way to achieve this goal is to make use of one of the modifications of the Hartree-Fock self-consistent field method. This problem is discussed in more detail in Chapter 28. 

For the case of a central field, the energy of an atom does not depend on magnetic quantum number m/. This means that the energy level, characterized by quantum numbers n and /, is degenerated 2/+1 times. For a pure Coulomb field there exists additional (hydrogenic) degeneration the energy of such an atom does not depend on /. Wave function (1.14) may... 

It is a reasonably good approximation to consider an alkali atom as a single electron moving in the modified Coulomb field of the ionic core, and this approximation has been made in almost all theoretical investigations of positron scattering by the alkali atoms. The interaction of the electron with the core is expressed as a local central potential of the general form... 

The simplest description of the effect of the central ion on the surrounding ligands reduces to that of the Coulomb field affecting the positions of the poles of the Green s... 

With such a model, the ion-ion interactions are obtained by calculating the most probable distribution of ions around any central ion and then evaluating the energy of the configuration. If (r) is the spherically symmetrical potential in the solution at a distance r from a central ion i of charge z 8, then (r) will be made up of two parts ZxZlDv the coulombic field due to the central ion, and an additional part, ai(r), due to the distribution of the other ions in the solution around t. The potentials at(r) and must satisfy Poisson s equation = — 47rp/D at every point... 

Excess Charge Density near the Central Ion Is Given by a Classical Law for the Distribution of Point Charges in a Coulombic Field... 

The ions of the pair together form an ionic dipole on which the net charge is zero. Within the ionic cloud, the locations of such uncharged ion pairs are completely random, since, being uncharged, they are not acted upon by the Coulombic field of the central ion. Furthermore, on the average, a certain fraction of the ions in the electrolytic solution will be stuck together in the form of ion pairs. This fraction will now be evaluated. 

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). 

We pass now to the spectra of the alkali atoms (fig. 12, Plate VI). They are considered to arise in this way an electron, the so-called radiating electron, moves in the fi.eld due to the nucleus and the rest of the electrons, and by itself causes the spectrum. This view is justified in the first place by the fact that in the alkalies, according to experiment, one electron is much more loosely bound than the rest, so that this electron is chiefly responsible for the chemical behaviour of the alkalies on the other hand, we shall find, in the discussion of the periodic system in next chapter, that the remaining Z — 1 electrons form so-called closed shells, round which the odd electron, i.e. the radiating electron, revolves. The field in which it moves is centrally symmetric, so that the potential depends only on the distance from the nucleus the Coulomb field of the nucleus is screened by the remaining Z — 1 electrons, and it is just these deviations from the Coulomb field which causes the differences between the spectra of the alkalies and that of hydrogen. 

For this reason we shall now deal with the motion of a particle in a central field of force. The motion in a Coulomb field of force (such as we have in the case of the hydrogen atom) will be found from this as a special case. 

The only assumption which we make is that the core (which includes one nucleus in the case of one atom and several in the case of a molecule) is small in comparison with the dimensions of the path of the radiating electron. The field will then closely resemble a Coulomb field over most of the path outside the core the distance of the aphelion from the centre point of the core will be determined only by the potential energy in the aphelion, it is therefore equal for all loops of the path independently of whether these loops are similar to one another (as for a central field) or not. Accordingly an effective quantum number n may be so defined that the relation... 

Central Coulomb Spin-Orbit Crystal Field... 

The third and most accurate method is that of the self-consistent field, due to Hartree(36). Here the interaction of the electrons is replaced by a central field, superposed upon the Coulomb field of the nucleus. The central field is chosen in such a way that the electrons, moving in the modified field of the nucleus according to wave mechanics, will just give rise to a charge distribution from which this central field would result on the basis of electrostatics. In contrast to the Thomas-Fermi distribution the last two methods have to be applied to every atom individually and are hence far more troublesome to handle. 

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