Atomic central field

Two coupled first order differential equations derived for the atomic central field problem within the relativistic framework are transformed to integral equations through the use of approximate Wentzel-Kramers-Brillouin solutions. It is shown that a finite charge density can be derived for a relativistic form of the Fermi-Thomas atomic model by appropriate attention to the boundary conditions. A numerical solution for the effective nuclear charge in the Xenon atom is calculated and fitted to a rational expression. 

S.F. Abdulnur et al., Atomic central-field models for open shells with application to transition metals. Phys. Rev. A 6, 889-898 (1972)... 

Notice that the density of a complete p, d, f, atomic subshell, or an incomplete sub shell in the central field approximation is rotationally invariant [58]. Thus only s-type charge density fitting functions are needed in any atomic central-field calculation. However if the central-field approximation is not invoked then very-high angular momenta are required to fit the density. From a practical point of view it might be better to set off center s-type fitting functions. 

One of the limitations of HF calculations is that they do not include electron correlation. This means that HF takes into account the average affect of electron repulsion, but not the explicit electron-electron interaction. Within HF theory the probability of finding an electron at some location around an atom is determined by the distance from the nucleus but not the distance to the other electrons as shown in Figure 3.1. This is not physically true, but it is the consequence of the central field approximation, which defines the HF method. 

The electrons within the atom are actually not quantised in parabolic coordinates, but instead, on account of the central field of the atom core, in polar co-ordinates. It would, then, not be logical to attempt to select favoured values of m and n3. Instead, we shall calculate the quantity... 

Added February 10, 1927.—J. H. Van Vleck in Proc. Nat. Acad. America, vol. 12, p. 662 (December, 1926), has discussed the mole refraction and the diamagnetic susceptibility of hydrogen-like atoms with the use of the wave mechanics, obtaining results identical with our equations (24) and (34). He also considered the effect of the relativity corrections (which is equivalent to the effect of a central field) and concluded that equation (24), derived by the use of parabolic instead of spherical co-ordinates, is not invalidated.]... 

Hartree, DR. The wave mechanics of an atom with a non-coulomb central field. 

Hartree, DR. Wave mechanics of an atom with a non-coulomb central field. HI. Term values and intensities in series in optical spectra. Proc Cambridge Phil Soc 1928 24 (Pt. 3) 426-37. 

Although Dirac s equation does not directly admit of a completely self-consistent single-particle interpretation, such an interpretation is physically acceptable and of practical use, provided the potential varies little over distances of the order of the Compton wavelength (h/mc) of the particle in question. It allows, for instance, first-order relativistic corrections to the spectrum of the hydrogen atom and to the core-level densities of many-electron atoms. The latter aspect is of special chemical importance. The required calculations are invariably numerical in nature and this eliminates the need to investigate central-field solutions in the same detail as for Schrodinger s equation. A brief outline suffices. 

Despite the complication due to the interdependence of orbital and spin angular momenta, the Dirac equation for a central field can be separated in spherical polar coordinates [63]. The energy eigenvalues for the hydrogen atom (V(r) = e2/r, in electrostatic units), are equivalent to the relativistic terms of the old quantum theory [64]... 

The solutions of the angular dependent part are the spherical harmonics, Y, known to most chemists as the mathematical expressions describing shapes of (hydrogenic) atomic orbitals. It is noted that Y is defined only in terms of a central field and not for atoms in molecules. 

At this point one question must be answered Is the potential calculated in the manner above path independent [21] Equivalently, is the field given by Equation 7.33 curl-free For one-dimensional cases and within the central field approximation for atoms, it is. For other systems, there is a small solenoidal component [21,22] and we will see later that it arises from the difference in the kinetic energy of the true system and the corresponding Kohn-Sham system (in this case the HF system and its Kohn-Sham counterpart). For the time being, we explore whether the physics of calculating the potential in the manner prescribed above is correct in the cases where the curl of the field vanishes. 

Given in Table 7.1 are the results [24] of the total energy of some atoms obtained by solving the Kohn-Sham equation self-consistently with the exchange potential Wx within the central field approximation. The energy is obtained from Equation 7.10... 

In our non-BO calculations performed so far, we have considered atomic systems with only -electrons and molecular systems with only a-electrons. The atomic non-BO calculations are much less complicated than the molecular calculations. After separation of the center-of-mass motion from the Hamiltonian and placing the atom nucleus in the center of the coordinate system, the internal Hamiltonian describes the motion of light pseudoelectrons in the central field on a positive charge (the charge of the nucleus) located in the origin of the internal coordinate system. Thus the basis functions in this case have to be able to accurately describe only the electronic correlation effect and the spherically symmetric distribution of the electrons around the central positive charge. 

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... 

These two cases are both cases of localization, in the sense that both are relative to an atomic central potential. However, one practically fills independently of the chemical potential pp, the other may be emptied easily by perturbing fields, or by thermal motion. 

Most atomic transitions are due to one electron changing its orbital. Using the central-field approximation, we have the angular part of the orbital function being a spherical harmonic, for which the selection rule is A/= 1 [(3.76)]. Hence for a one-electron atomic transition, the / value of the electron making the jump changes by 1. 

Let us first review the different atomic orbitals. These orbitals are obtained by solving the Schrodinger equation for a particle of mass m, the electron, in a central field produced by the nucleus ... 

In the present paper we give formulas for S for all AO pairs involving ns, npa, and npr AO s for = 1, 2,3, and 5 using Slater AO s. We also give numerical tables for the most important of these cases, applicable to a wide variety of atom pairs over a wide range of R. In addition, we show how the tables, though based on Slater AO s, may be used to obtain S values for central-field AO s, as well as for hybrid AO s. Explicit tables of hybrid S values are also given for n 2. 

Although the one-dimensional model bears little resemblance to any real molecular system, many of its features carry over to cases of practical interest. Suppose we consider three-dimensional motion in a central field as in atoms. The orbitals or single-electron functions now become atomic orbitals and can be classified in the usual manner as Is, 2a,. . . 2p, 3p,. . . 3d,. Suppose we are dealing with an atom in which there are two electrons of the same spin (a, say) occupying the 2a and 2p orbitals (inner shells being ignored for the present). Then the antisymmetric product function 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... 

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. 

Thus, in the central field approximation the wave function of the stationary state of an electron in an atom will be the eigenfunction of the operators of total energy, angular and spin momenta squared and one of their projections. These operators will form the full set of commuting operators and the corresponding stationary state of an atomic electron will be characterized by total energy E, quantum numbers of orbital l and spin s momenta as well as by one of their projections. 

---