Central-field approximation

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. 

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. 

I assume that in nuclei the nucleons may. as a first approximation, he described as occupying localized 1. orbitals to form small clusters. These small clusters, called spherons. arc usually hclions, tritons, and dincutrons in nuclei containing an odd number of neutrons, an Hc i cluster or a deuteron may serve as a spheron. The localized l.v orbitals may be described as hybrids of the central-field orbitals of the shell model. 

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... 

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. 

Central field approximation, angular momentum and spherical... 

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. 

Zero-order energy of the central field approximation, described by the central symmetric part of the potential, does not contain interaction of the momenta. Therefore, in zero-order approximation all states of a given configuration differing from each other by quantum numbers m, m, i.e. by different orientation of orbital and spin momenta 1, and s,-, have the same energy, and the corresponding level is degenerated (4/ + 2) times. 

The group of rotations of a three-dimensional space stands apart in atomic spectroscopy. This is mostly due to the high accuracy of the central field approximation, on which the entire modem theory of complex atoms and ions is based. 

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... 

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. 

In the central field approximation, when radial wave functions not depending on term are usually employed, the line strengths of any transition may be represented as a product of one radial integral and of a number of 3n./-coefficients, one-electron submatrix elements of standard operators (C(fc) and/or L(1 S(1)), CFP (if the number of electrons in open shells changes) and appropriate algebraic multipliers. It is usually assumed that the radial integral does not depend on the quantum numbers of the vec-... 

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. 

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. 

Let the laser field be expressed as s(t) — o(t) sin(coLt), where o(t) is the pulse envelope function, including polarization, and col is the central frequency. For simplicity, consider a 8 function excitation. In the rotating wave approximation, the field is expressed as e(t) = S(t) exp(—icoLt) with field strength sq. Integration over t in Eq. (25) gives... 

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. 

Not even the SCF procedure can overcome this problem. In the case of atoms, the central field remains a valid and good approximation. Assuming a rigid linear structure in the molecular case is clearly not good enough, although it contains an element of truth. This inherent problem plagues all LCAO-SCF calculations to an even more serious extent. 

It is worth emphasizing two aspects of eq. (19) expressing minimum requirements for a plausible L.C.A.O. approximation to the partly filled shell, and which have been neglected recently by several authors. One is the introduction of the central-field covalency in the form of the variable... 

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