Field Model of Tetrahedral and Octahedral Molecules

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. 

we have in this model a nuclear potential energy Vx(r) given by 

Here the central nucleus carries charge ze (for methane, z— 6) and the total surface charge spread uniformly on the sphere of radius R is ne (n=4 for methane). 

Since the nuclear framework is spherical according to equation (87), we can immediately use the TF equation (10) to describe the self-consistent field, provided that  

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 

---

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. 

---