Central field model

In Thomas-Fermi theory, adoption of the simple central field model for neutral molecules at equihbrium leads to simple energy relations—well supported by SCF calculations [12]—such as = (Vne + 2Vnn)- Evidently, nothing of the like apphes to vaience> but wc may well inquire how things are with... 

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

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... 

Central Field Model of Tetrahedral and Octahedral Molecules.—The idea is very simple, and has long been exploited in the sense of one-centre expansions of molecular wave functions in a molecule like CH4. However, to exemplify the way the density description can afford answers to questions (i)—(iii) above, we take the model literally in which, in methane for example, we smear the protons uniformly over the surface of a sphere of radius R, equal, in the methane example to the C—H bond length. 

Without the need to go into more quantitative detail, which is given fully elsewhere,35 there are two immediate consequences of the TF solution for this central field model of molecules namely that the chemical potential for the neutral molecules under discussion is identically zero, and secondly there is an equilibrium bond length R=Re say, which is specified by... 

Figure 3 Equilibrium bond length in density description of central field model of tetrahedral molecules.

The above treatment of the central field model was based on the TF Euler equation (24), applied self-consistently with Fn (r) which was given by equation (87). But as for atoms, we have generally, by multiplying equation (24) by the density and integrating over all space... 

Adoption of Central Field Model at Equilibrium.—To make progress, we now adopt the central field model, for which, for neutral molecules, we saw that the chemical potential n was zero. Secondly, we established the existence of an equilibrium bond length at Re, given by equation (89), where the virial theorem for equilibrium under purely Coulomb forces takes the usual form... 

This simple relation will be confronted with the results of accurate self-consistent calculations for a variety of small molecules in the next section, where we shall see that, in spite of its derivation from the central field model above, it is valid in a much wider context, to a useful approximation. 

Thus, although equations (96) and (98) were derived above by working out the density description for the central field model of tetrahedral and octahedral molecules, Figures 4 and 5 confirm the validity of these relations for a wide variety of molecules, using self-consistent wave function calculations. This is the more remarkable because the simplest TF density description is ensured, as a statistical theory, to become asymptotically valid for large numbers of electrons N, whereas the results of Figures 4 and 5 are for molecules with 24 (c/. Appendix 1). But in view of this last point, it is obviously important to study molecules with a larger number of electrons. This leads us back to the tetrahedral and octahedral molecules. 

The fact that in the tetrahedral and octahedral molecules the bond length at equilibrium, in the TF density description of the central field model, goes as R z-Vs in equation (92) is reflected in the fact that the model gives Vaa too large a value, as a comparison of the magnitudes of Vnn in Figures 5 and 6 shows. In tetrahedral and octahedral molecules, equations (91) and (100) show that the relation is more like i eoc AT1/3. 

March and Parr also point out as already mentioned above that the interpretation of the result ju=0 for the neutral Thomas-Fermi atom (cf the result for the central field model for molecules in Section 9, where the chemical potential is also zero for neutral molecules) is that the gross trend with Z is juocZ-1/3 at large Z. Of course, such a discussion would have to be refined considerably to reproduce the chemically important periodic effects in p, which will be focused upon below. 

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... 

In the present chapter, we describe how the concept of closed shells arises within the central field model, and what consequences flow from it. 

We thus have a simple model (the aufbau or building-up principle of Bohr [1] and Stoner [2]) which correctly predicts the periodic structure of Mendeleev s table of the elements. More precisely, one should state that Mendeleev s table is the experimental evidence which allows us to use an independent electron central field model and to associate each electron in a closed shell with a spherical harmonic of given n and i, because there is no physical reason why a particular l for an individual electron should be a valid quantum number angular momentum in classical mechanics is only conserved when there is spherical symmetry. 

Within the central field model, it is useful to define for each value of i an effective radial potential... 

Again, it is assumed that the reader is familiar with the usual content of an undergraduate course in atomic physics, viz. the principles of quantum mechanics, the hydrogen atom, elementary treatments of angular momentum, of spin-orbit interaction, the Pauli principle, static mean fields, the central field model, the building-up principle, spectroscopic notation and the working of dipole selection rules. 

In this expansion V now contains the electrostatic deviation from the central field model Ves and the additional small hyperfine perturbation h ... 

The molecular orbital (MO) is the basic concept in contemporary quantum chemistry. " It is used to describe the electronic structure of molecular systems in almost all models, ranging from simple Hiickel theory to the most advanced multiconfigurational treatments. Only in valence bond (VB) theory is it not used. Here, polarized atomic orbitals are instead the basic feature. One might ask why MOs have become the key concept in molecular electronic structure theory. There are several reasons, but the most important is most likely the computational advantages of MO theory compared to the alternative VB approach. The first quantum mechanical calculation on a molecule was the Heitler-London study of H2 and this was the start of VB theory. It was found, however, that this approach led to complex structures of the wave funetion when applied to many-electron systems and the mainstream of quantum ehemistry was to take another route, based on the success of the central-field model for atoms introduced by by Hartree in 1928 and developed into what we today know as the Hartree-Foek (HF) method, by Fock, Slater, and co-workers (see Ref. 5 for a review of the HF method for atoms). It was found in these calculations of atomic orbitals that a surprisingly accurate description of the electronic structure could be achieved by assuming that the electrons move independently of each other in the mean field created by the electron cloud. Some correlation was introduced between electrons with... 

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