Central-field approximation for

Funabashi, K., Magee, J. L., J. Chem. Phys. 26, 407, Central field approximation for the electronic wave functions of simple molecules/ Interaction of all electrons included, but no configurational interaction. 

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. 

We shall confine ourselves to the case of one shell of equivalent electrons. In the central field approximation, for LS coupling the states of the lN configuration are characterized by orbital and spin momenta. In fact, using (13.15), (5.15) and the condition... 

This work W,ie(r) is path-independent for the symmetrical density systems noted previously since the V x = 0 for these cases. It is important to note, however, that the corresponding Fermi-Coulomb hole charge distribution Pxc(r r ) which gives rise to the field need not possess the same symmetry for arbitrary electron position. For example, in either closed shell atoms or open-shell atoms in the central-field approximation for which the density is spherically symmetric, the Fermi-Coulomb hole is not, the only exception being when the electron is at the nucleus. 

In order to solve this equation we use the central field approximation for which the following assumptions are made ... 

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... 

The basic Hamiltonian for the central field approximation is thus ... 

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. 

The operator of the energy of electrostatic interaction of electrons in (14.65) is represented as a sum of second-quantization operators, and the appropriate submatrix element of each term is proportional to the energy of electrostatic interaction of a pair of equivalent electrons with orbital Lu and spin S12 angular momenta. The values of these submatrix elements are different for different pairing states, since, as follows from (14.66), the two-electron submatrix elements concerned are explicitly dependent on L12, and, hence, implicitly - on S12 (sum L12 + S12 is even). It is in this way that, in the second-quantization representation for the lN configuration, the dependence of the energy of electrostatic interaction on the angles between the particles shows up. This dependence violates the central field approximation. 

The monograph of Condon and Shortley [2] was a major work of reference for a whole generation of spectroscopists [20]. It treats an atom in the central-field approximation and it does not require deep knowledge of the theory of groups. 

This work done is path-independent since V x R(r) = 0. For systems of certain symmetry such as closed shell atoms or open-shell atoms in the central-field approximation, the jellium and structureless-pseudopotential models of a metal surface considered here, etc., the work Wxc (r) and Wt (r) are separately path-independent since for these cases Vx xc(r) = VxZt (r) = 0. 

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. 

Harrison has performed SIC-LSD calculations for the 3d series but without sphericalizing the spin-orbital densities. The SIC is calculated for Cartesian orbitals and then the central-field approximation is reintroduced by spherically averaging the contributions to the energy and to the potential arising from a given shell. As Harrison pointed out, the errors introduced by sphericalizing the orbital densities are present even for spherical atoms because of the non-linear orbital density dependence of the SIC (see also Ref. 186). His results for the d" s-d" s separation are shown in Fig. 7. The... 

What makes eq. (1) difficult to solve is the presence of the Vee two-electron operator. Without it, the solution would be straightforward, as is the case for all hydrogen-like systems where we have one electron under the influence of a central field. Hydrogen-like systems are the only cases where analytic solutions are known in closed form. These solutions, or functions very similar to them, are the most natural basis to use in seeking the solutions for systems with more than two electrons. One approach is to treat one electron at a time and solve for this particular electron when it is under the average field of the others and nuclei. This is known as the central field approximation, and is the basis for the treatment of larger systems. 

The central field approximation and the simplifications which result from it allow one to construct a highly successful quantum-mechanical model for the AT-electron atom, by using Hartree s principle of the self-consistent field (SCF). In this method, one equation is obtained for each radial function, and the system is solved iteratively until convergence is obtained, which leaves the total energy stationary with respect to variations of all the functions (the variational principle ). The Hartree-Fock equations for an AT-electron system are equivalent to several one electron radial Schrodinger equations (see equation (2.2)), with terms which make the solution for one orbital dependent on all the others. In essence, the full AT-electron problem is approximated by a smaller number of coupled one-electron problems. This scheme is sometimes (somewhat inappropriately) referred to as a one-electron model in fact, the Hartree-Fock equations are a genuine AT-electron theory, but describe an independent particle system. 

The system of equations (1.8) is based on the central field approximation, and therefore its application to real atoms is entirely dependent on the existence of closed shells, which restore spherical symmetry in each successive row of the periodic table. For spherically symmetric atoms with closed shells, the Hartree-Fock equations do not involve neglecting noncentral electrostatic interactions and are therefore said to apply exactly. This does not mean that they are expected to yield exact values for the experimental energies, but merely that they will apply better than for atoms which are not centrally symmetric. One should bear in mind that, in any real atom, there are many excited configurations, which mix in even with the ground state and which are not spherically symmetric. Even if one could include all of them in a Hartree-Fock multiconfigura-tional calculation, they would not be exactly represented. Consequently, there is no such thing as an exact solution for any many-electron atom, even under the most favourable assumptions of spherical symmetry. 

In the early days of atomic theory, it was often assumed that only an empirical understanding could be achieved beyond the first few rows of the periodic table, where the simple formulation of the aufbau principle breaks down. The modern view is that shell and subshell filling can be accounted for within the central field approximation, provided the centrifugal barrier effects are included. 

Hyperspherical methods have the merit of providing a dynamical picture of double excitation and double escape for which the central field approximation is inappropriate. Initially, very accurate calculations were not achieved in this way, and so the hyperspherical method was mainly used as a framework to understand the results obtained by other methods. This situation was transformed by the work of Tang and Shimamura [330] who have performed the most detailed calculations to date on systems with two electrons. 

As long as n and remain good quantum numbers, the independent particle model and the central field approximation both apply, and quantum chaos does not arise. We can thus identify two situations where chaos could emerge the first is a complete breakdown in the independent electron approximation (due, for example, to strong correlations) and the second is a distortion of the central field approximation (due, for example, to a strong external field). 

Within the central field approximation one obtains from Eq. 18 for an one-electron atom the following radial equation ... 

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