Lower energy doubly occupied orbital

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. 

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... 

There is a second interaction that involves the empty antibonding 7t orbital and the metal d orbital with the same symmetry xz), which is lower in energy than the n orbital (3-37b). If this latter orbital is doubly occupied, this interaction is stabilizing, and it leads to a transfer of electron density from the metal to the ligand. This is therefore a hack-donation interaction, where ethylene plays the role of a itt acceptor, using its empty tt orbital. The doubly occupied orbital, mainly concentrated on the metal, is part of the d block of the complex it can be described as a metal d orbital that is stabilized by a bonding interaction with the n orbital on ethylene. 

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. 

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. 

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... 

When discussing the electronic structure of molecules and solids, one-electron descriptions, such as the molecular orbitals of Equation 8.1, are quite intuitive. It is common to talk about individual electrons occupying particular states. For example, reactions often occur by the mixing of the highest occupied molecular orbital (HOMO) of one species and the lowest unoccupied molecular orbital (LUMO) of another. In such a reaction the electrons in the HOMO state move into the new mixed orbital, lowering their energy. The HOMO and LUMO states are each pairs of one-electron molecular orbitals, since in the simplest case an orbital giving the spahal distribution for a spin up electron has an identical partner for spin down. Mulh-electron wavefunctions that describe the whole electronic structure in this picture are constructed from the one-electron states. So, for example, in a four-electron system in which all the electronic states are doubly occupied (spin up and spin down), based on Hartree-Fock theory we can write ... 

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