Lower energy doubly occupied

The total it electron energy is the sum of occupied orbital energies multiplied by two if. as is usually the ease, the orbital is doubly occupied. The charge densities and free valency indices were treated in separate sections above. The bond order output should be read as a lower triangular serni matrix. The bond order semi matrix for the butadiene output is shown in Fig. 7-7. 

In the ground state, the M = N/2) states in the lower-half of the bulk band are doubly occupied, so the total reduced energy for 0k — 0k — 8k) is... 

These both have the same energy so that, in the absence of other determining factors, the electrons go one into each with the same spin (or an equivalent state). This means that they are kept apart by the antisymmetry principle and so the energy Is lowered by the reduction of Coulomb repulsion. In this rather exceptional case, therefore, the orbitals are not all doubly occupied and we cannot carry out any simple transformation into localised orbitals. 

In an ESR study of 1,1,3,3-difluoroallyl radicals, Krusic and coworkers were able to demonstrate that the barrier to rotation of such apparently planar radicals is substantially reduced [18]. Although allyl itself has a rotational barrier of 15 kcal/mol [19, 20], 1,1,3,3-tetrafluoroallyl, 1, had a barrier of but 7.2 kcal/mol. The observed 19F hfs constants (42.6 and 39.7 G) were consistent with 1 being a planar system. It is likely that the lowering of the rotational barrier of 1 derives from a destabilizing interaction between the fluorine lone pairs and the doubly-occupied allyl tt-MO which diminishes the net allylic resonance energy, as well as from stabilization of the transition state due to pyramidalization. 

A donor substituent may be represented by a doubly occupied orbital D, at (a + 2/3) for O and at (a + 1.5/ ) for N. Hence D always lies lower than the HOMOs of the diene and the dienophile. Scheme c illustrates the tricky case involving a first-order D-rc interaction and a second-order D- 2 term. One may wonder if the HOMO of the substituted dienophile cannot be higher than that of the diene. In fact, we just need to take a double bond for D to see that the HOMO of hexatriene lies higher than the butadiene HOMO when n is raised to the level of 2, the latter rises further, so that the diene always has the higher energy. 

The problem of the anion is slightly more complicated (Figure 7.6). The p orbital is doubly occupied, so four interactions must be considered the two-electron stabilizing (2) and (4) and the four-electron destabilizing interactions (1) and (3). Fortunately, their effects are complementary. 7t Mi. and 7i Me lie at lower energy than nMe and 7i Me, so... 

In order to see if it is possible to neutralise this effect of the a-system we performed a second calculation which used localised orbitals for the a-system as well as for the Tt-system. In this calculation one perfect-pairing structure was used for the C-C bonds of the a-system. All orbitals were localised on the C-H fragments. Doubly occupied orbitals were used for the C-H bonds, and strictly localised singly occupied orbitals for the C-C bonds. This calculation again yields a rectangular geometry with a much lower resonance energy. The bond... 

Li20 has also been studied by the separated-electron pair method.422 A STO basis set of DZ quality was used. Doubly occupied geminals were used in constructing the wavefunction, and hence electron correlation is included. The molecule was again predicted to be linear. The best energy was 0.15 hartree lower than in the above SCF calculations. A detailed analysis of the MO s was given in this paper. 

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