The central-field problem

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 central field problem distinguishes celestial mechanics from other areas of classical dynamics. This deals with the motion of a test particle, whose mass is negligible with respect to the central body, in the gravitational field of a point mass. The extended version of this problem is to allow the central mass to have a finite spatial extent, to depart from spherical symmetry, and perhaps to rotate. The basic Newtonian problem is the foUowing. 

Here the solution to the central field problem makes specific use of the gravitational force. The substitution of... 

The method of superposition of configurations is essentially based on the assumption that the basic orbitals form a complete set. The most popular basis used so far in the literature is certainly formed by the hydrogen-like functions, which set contains a discrete and a continuous part. The discrete subset corresponds physically to the bound states of an electron around a proton, whereas the continuous part corresponds to a free electron scattered by a proton, or classically to the elliptic and hyperbolic orbits, respectively, in a central-field problem. 

The selection rules (3.76)—(3.78) were derived without reference to the form of the radial functions hence they are valid for any one-particle central-field problem. For example, they hold well for the sodium valence electron, which moves outside a closed-shell structure. 

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. 

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. 

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. 

All these questions, as we shall see, can be discussed fruitfully from the density description of molecules. But because, as we have already emphasized, the multicentre problem is difficult to tackle even in the simplest TF density description, we shall attempt to tackle questions (i)—(iii) above by turning immediately to a central field model which was solved by March86 in the TF density description. The model was set up with tetrahedral and octahedral molecules in mind, for example GeH4, UF6 etc. It has been used recently by Mucci and March88 in a discussion of energy relations for molecules at equilibrium. We shall summarize their main results below, after discussing the solution of the central field model.85... 

Because of the problem associated with Teller s theorem, discussed in Section 11, let us again examine the predictions of the central field model of molecules of Sections 9 and 10. From this model stemmed the energy relations (96)—(98). Equation (81) is again the complete expression for the sum of the eigenvalues in this simplest density description. Using equation (93), with the chemical potential equal to zero, as was demonstrated to be so for neutral molecules in the central field model, one can eliminate Fen + 2Fee by subtracting equations (81) and (93), to obtain... 

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 field H = Cl+(3,0) of the Hamilton quaternions and the ring 0(3,0) of the Clifford biquaternions are relevant of the general theory of the Clifford algebra C1(.E) = Cl(p, n — p) associated with an euclidean space E = Rp,n p. They correspond to the initial construction of the Clifford algebras. Especially, the field of the Hamilton quaternions plays an important role in the solution of the central potential problem. 

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 equations for the two radial wave functions g and f in the Dirac central field problem are [9.9,10]... 

Initially, both the Hartree method and the Hartree-Fock method were applied exclusively to atoms, where the spherical symmetry of the system allowed one to simplify the problem considerably. These approximate methods were (and still are) often used together with the central field approximation, to enforce the condition that electrons in the same shell have the same radial part, and to restrict the variational solution to be a spin eigenfunction. 

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