Potential fields spherically symmetrical

In order to improve the theoretical description of a many-body system one has to take into consideration the so-called correlation effects, i.e. to deal with the problem of accounting for the departures from the simple independent particle model, in which the electrons are assumed to move independently of each other in an average field due to the atomic nucleus and the other electrons. Making an additional assumption that this average potential is spherically symmetric we arrive at the central field concept (Hartree-Fock model), which forms the basis of the atomic shell structure and the chemical regularity of the elements. Of course, relativistic effects must also be accounted for as corrections, if they are small, or already at the very beginning starting with the relativistic Hamiltonian and relativistic wave functions. 

Spin-Orbit Interaction. If an electron of mass m and orbital angular momentum L moves with a velocity v in an electric field (internal to the atom or molecule) Eint, or in an electric potential 0int(r) (due to the nucleus), then it experiences, in addition to Eint, a magnetic field Bint. If the potential is spherically symmetric, then the electric field is simply... 

We derive approximate mobility formulas for the simple but important case where the double-layer potential remains spherically symmetrical in the presence of the applied electric field (the relaxation effect is neglected). In this case, it can be shown that Eq. (22.10) tends to, in the limit k oo. 

For a Dirac particle in a central field 4> is spherically symmetrical and A = 0. Setting the potential energy V(r) — e(r), the Dirac Hamiltonian becomes... 

Exact solutions of the Schrddinger equation are, of course, impossible for atoms containing 90 electrons and more. The most common approximation used for solving Schrddinger s equation for heavy atoms is a Hartree-Fock or central field approximation. In this approximation, the individual electrostatic repulsion between the electron i and the N-1 others is replaced by a mean central field giving rise to a spherically symmetric potential... 

Because of the unknown characteristic of nuclear forces, many different suppositions have been made, using available experimental evidence, regarding the nature of the potential energy V of a nuclear particle as a function of its position in the field of a nucleus, or of another nuclear particle. To a first approximation, the nuclear potential is assumed to be spherically symmetric, such as V is a function only of the distance r from the center of the field, thus being the same in all directions, and is representable by a curve as m Pig. 1 curves (a) to (f). 

Since the fictitious particle moves in a central force field described by a spherically symmetric potential function U(r), its angular momentum is conserved. Therefore, the motion of the fictitious particle will be in a plane defined by the velocity and the radius vectors. The Lagrangian may then be conveniently expressed in polar coordinates as... 

In the absence of external fields, we may take axes that move laterally with the molecule, thus eliminating translational motion. In these co-ordinates a diatomic molecule becomes equivalent to a single particle with mass ti=MaMjs/(Ma+Mb), moving in a spherically symmetrical potential U(R), where R is the intranuclear separation. The Schrodinger equation is therefore... 

We now specialize the discussion to the ligand field theory situation and define the orthonormal set of spin-orbitals we shall use in the determinantal expansion of the many-electron functions Vyy for the groups M and L. First we suppose that we have a set of k orbitals describing the one-electron states in the metal atom these will be orthonormal solutions of a Schrodinger equation for a spherically symmetric potential, V<,(r), which may be thought of as the average potential about the metal atom which an electron experiences ... 

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... 

Using a central field approximation in which it is assumed that each electron moves independently in an average spherically symmetric potential, it is possible to solve for the energies of the different configurations. Calculations of this type show that the / -configuration is the lowest energy configuration for the trivalent lanthanides and actinides. 

In the virial equation as given by Eq. (3.12), the first term on the right is unity, and by itself provides the ideal-gas value for Z. The remaining terms provide corrections to the ideal-gas value, and of these the term B/ V is the most important. As the two-body-interaction term, it is evidently related to the pair-potential function discussed in the preceding section. For spherically symmetric intermolecularforce fields, statistical mechanics provides an exact expression relating the second virial coefficient B to the pair-potentialfunctionW() ) ... 

Even-even nuclei may be described by a spherically symmetric Coulomb potential Vext of an extended nucleus with charge number Z. Pure QED effects due to the interaction with the free radiation field are carried by the interaction Hamiltonian... 

In spherically symmetric systems the induced diamagnetism depends primarily on the mean square radius of the valence electrons as the small contribution from the inner-shell electron core can usually be neglected 1 ). In the case of molecules with symmetry lower than cubic, the quantum mechanical treatment by Van Vleck 23> indicates that another term must be added to the Larmor-Langevin expression in order to calculate correctly diamagnetic susceptibilities. This second term arises because the electrons now suffer a resistance to precession in certain directions due to the deviations of the atomic potential from centric symmetry. The induced moment will now be dependent on the orientation of the molecule in the applied magnetic field and thus in general the diamagnetic susceptibility will not be an isotropic quantity 19-a8>. 

Conservation of angular momentum may be applied to more general systems than the one described here. It is at once evident that we have not used the special form of the potential-energy expression except for the fact that it is independent of direction, since this function enters into the r equation only. Therefore the above results are true for a particle moving in any spherically symmetric potential field. 

Angular momenta can likewise be expressed in this manner. Thus, for one particle in a spherically symmetric potential field, the angular momentum about the z axis was defined in Section le by the expression... 

This integral (aside from the factor Ze2/32ir2ao) represents the mutual electrostatic energy of two spherically symmetrical distributions of electricity, with density functions e", i and respectively. It can be evaluated by calculating the potential due to the first distribution, by integrating over dri, and then evaluating the energy of the second distribution in the field of the first. 

The field due to an ion and its ionic atmosphere. The ionic atmosphere is spherically symmetrical and the potential drops off non-linearly with distance, and so the field is non-uniform (see Figure 11.2(b)). 

A particle with charge e and anomalous moment /Xa subjected to a spherically symmetric electric field E(x) = would be described by the potential... 

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