The Atomic Central Field Problem

Some of the consequences of the spherical symmetry of the hamiltonian of an atomic system for the Green s function are explored in this section. A number of results concerning tensor operators are invoked and the reader are assumed to be familiar with the Wigner 3j and 6j symbols. 

The creation operator is a tensor operator of rank I with respect to the orbital angular momentum and of rank with respect to spin. For instance. 

Similar expressions can be derived for the annihilators It must be recog- 

Matrix elements of the field operator (0 over the many-electron states can 

Since only the reduced matrix element and the radial function depend on the label n, it is advantageous to define a more general reduced matrix element 

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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. 

For D = 3, and putting zq = z in Eq. (24) to obtain the Slater sum S, use of the explicit form of V in Eq. (22) readily allows one to verify that the diagonal form of Eq. (24) is indeed an exact solution of Eq. (25). Later, Amovilli and March [20] made similar progress on central field problems. It remains of interest to treat atoms in intense electric fields by direct use of the Slater sum rather than by use of the off-diagonal canonical density matrix. 

As to future directions, the problem of the canonical density matrix, or equivalently the Feynman propagator, for hydrogen-like atoms in intense external fields remain an unsolved problem of major interest. Not unrelated, differential equations for the diagonal element of the canonical density matrix, the important Slater sum, are going to be worthy of further research, some progress having already been made in (a) intense electric fields and (b) in central field problems. Finally, further analytical work on semiclassical time-dependent theory seems of considerable interest for the future. 

The hydrogen atom is a typical case of the central-field problem. As was shown in Fig. 19.5, the proton is at the center with a charge + e while the electron is at a distance r with a charge —e. The coulombic force acts along the line of centers and corresponds to a potential energy, V(r) = —e /4ncQr. 

The selection rules for the different types of polarization are, of course, significant only when there is a unique z direction, due, for example, to the presence of a uniform magnetic field. This subject will be discussed more fully in the following chapter, where the Zeeman effect is considered. It is apparent from the derivation of the above selection rules that they are not limited to the hydrogen atom but are valid for any central field problem where the angular portion of the wave function is identical with that of the hydrogen atom. 

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. 

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. 

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. 

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