Touch-down energy

A useful mnemonic device for quickly setting down the HMOs for cyclic systems is Frosts circle.If a regular polygon of n sides is inscribed in a circle of diameter 4/3 with one comer at the lowest point, the points at which the comers of the polygon touch the circle define the energy levels. The energy levels obtained for benzene and cyclobutadiene with Frost s circle are shown in Fig. 1.12. 

One possible way to achieve nuclear fusion is to use magnetic fields to confine the reactant nuclei and prevent them from touching the walls of the container, where they would quickly slow down below the velocity required for fusion. Using 400-ton magnets, it is possible to sustain the reaction for a fraction of a second. To achieve a net evolution of energy, this time must be extended to about one second. A practical fusion reactor would have to produce 20 times as much energy as it consumes. Optimists predict that this goal may be reached in 50 years. 

The shape of the polygon being analyzed (e.g., a hexagon for benzene) is drawn in a ring with one vertex down, and the relative energies of the molecular orbitals are indicated by the vertices of the polygon that touch the circle. 

Fig. 1. Band structures of (a) CeRhj and (b) CePdj calculated using a fully relativistic LAPW method. Note how the narrow f bands in CeRhj (about 0.05 Ry above the Fermi energy) pull down to touch the Fermi energy in CePdj which would be consistent with a tetravalent-mixed valent picture of alloys between these materials. However, it should be noted that CeRhj has very nearly a full f electron occupied through hybridization.

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