Density description

Obviously, the density description suggests that homoantiaromatic molecules prefer Mobius 4q + 2 electron systems rather than Htickel 4q systems. This is in line with the PMO analyses of Hehre66,69 and Jorgensen72-73 (see Section III.C). 

In a nutshell (DFT methods) Electron density description of chemical phenomena, straightforward interpretation Large molecular systems can be treated Variable accuracy, validation necessary Little basis set dependence No systematic improvements. 

In the latter way of looking at the build-up of molecules, the successive addition of electrons to a positively charged system is reminiscent of the manner in which the atoms of the Periodic Table were considered in Chapter 1. Here again there are certain configurations permitted the electron clouds, and these cloud shapes (or probability density functions) can be described using quantum numbers. Such probability density descriptions are called molecular orbitals in analogy to the much simpler atomic orbitals. Although the initial setup and subsequent mathematical treatment for molecules are much more complicated than for atoms, there arise certain similarities between the two types of orbitals. 

However, for large numbers of electrons, a different approach is clearly required. This is afforded by the density description. Instead of a wave function with 3N spatial co-ordinates, one works with the ground-state electron density p(r), where this is explicitly the number of electrons per unit volume at position r. This is evidently a three-dimensional quantity, independent of the number of electrons, and is therefore a favourable tool for really large molecules. 

Total Energy for Heavy Neutral Atoms.—The fact that the neutral atom solution has the form I in Figure 1 implies that (x) - 0 as x - oo, in fact as 144/x which is readily verified to be an exact solution of the dimensionless TF equation (10), not however satisfying the atomic boundary condition (11). Since V(r) - 0 at infinity, it follows from equation (7) that, for this neutral case with N=Z, we must have p=0. The condition that, in the simplest density description of neutral atoms, the chemical potential is zero is important for the arguments which follow. We shall see below that one of the objectives of more sophisticated density descriptions must be to find p. Equations (25) and (26) can be rewritten in the form, using E= — T and p=0,... 

Thus,/0(A7Z) in equation (42) can be calculated from the known solutions of equation (10), the form of f0(N/Z), taken from the work of March and White,7 being plotted in Figure 2. Since the density description focuses so directly on E(Z, N), as in equation (42), it is natural that we should bring the result (42) of this simplest (TF) density theory into contact with the 1/Z expansion of E(Z, N) for atomic ions. That these two treatments are very intimately related... 

Before discussing the consequences of equation (48) for the total energy of positive ions, it is clearly of importance to understand how the density description has to be generalized beyond the TF approximation to account for the terms 0(Z2) and 0(Z5/3) in equation (48). 

We want to emphasize that in writing equation (51) we are still working at the level of a single Slater determinant, no electron correlation therefore being embodied as yet in the density description. The formal relaxation of this final restriction will be carried out in Section 15 below. 

In summary, equation (48) represents both a natural generalization of the simplest density description (TF approximation) and a valuable rearrangement of the Layzer 1/Z expansion. In principle, equation (43) and its rearrangement and partial sum (48) include many-electron correlation effects. However, the numbers /i(l) = i and /a(l) = -0.266 do not have any correlation included it may be that this enters only in the higher-order terms in equation (48) but this has not presently been established. We shall later discuss the inclusion of correlation in the density description. 

The density description focused attention on the total ionic energy E(Z, N) and led to the Z-1/3 expansion (48), when combined with the 1/Z series (43). Two further developments of E(Z, N) will be recorded here, following the work... 

Having discussed the basic equations of the density description and their application to atomic ions we turn now to the much more difficult problem of molecules. Even the simplest density description afforded by the TF theory presents severe computational problems for multicentre problems, as well as some conceptual difficulties on which we shall attempt to throw light in the ensuing discussion. 

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. 

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

Though, as emphasized, we must not expect equation (92) to be realistic for tetrahedral and octahedral molecules with heavy atoms in the outer positions, its form is of some interest. However, we will enquire, within this density description, what the answer to question (i), namely that the chemical potential is zero in this treatment, has to say about questions (ii) and (iii). We tackle the question of the energy relations in the next section. 

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 relations (96)—(98) were established on the basis of the simplest density description, the TF theory. We mentioned earlier a conceptual difficulty that... 

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 Parr12 also consider the chemical potential in the same limit. They argue that the meaning of = 0 in the Euler equation of the density description is that Np in this equation is a smaller-order term in the number of electrons than the other energy components. Thus gross features, of the kind exhibited in the energy relations (96)—(98), can be treated but the chemical potential and the nuclear-nuclear potential energy, require special care. 

Notwithstanding this, the above considerations suggest that a more refined theory, motivated by the density description, may be possible for some mole-... 

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