Orbital labeling system

We have so far said little about the nature ofthe space function, S. Earlier we implied that it might be an orbital product, but this was not really necessary in our general work analyzing the effects of the antisymmetrizer and the spin eigenfunction. We shall now be specific and assume that S is a product of orbitals. There are many ways that a product of orbitals could be arranged, and, indeed, there are many of these for which the application of the would produce zero. The partition corresponding to the spin eigenfunction had at most two rows, and we have seen that the appropriate ones for the spatial functions have at most two columns. Let us illustrate these considerations with a system of five electrons in a doublet state, and assume that we have five different (linearly independent) orbitals, which we label a, b,c,d, and e. We can draw two tableaux, one with the particle labels and one with the orbital labels. 

First, let us see how VB theory explains the symmetry properties of 4n versus An + 2 electronic systems in neutral rings. Consider a ring involving 2N atomic p orbitals, labeled from 1 to 2N, (3 in Scheme 5.1) and having one... 

Before we set up the model Hamiltonian for electrochemical electron transfer, we have to specify the models for the various parts of the system. For the electrons in the metal, we use the quasi-free electron model in which the electronic states are labeled by their quasi-momentum k. For outer-sphere electron transfer on metals, it is usually permissible to ignore the spin index - keeping it would introduce an additional factor of two, which can be incorporated into the interaction constants. On the reactant, we consider a single orbital, labeled a, with which the electrons are exchanged. 

Figure 7.29 The effect of hybridization on the MO diagram of A2 molecules. The orbital labels of the a hybrids are referred to using the same numbering system as for the nonhybrid case, i.e. starting from Sag as the lowest energy state they are numbered in order of energy.

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

Atoms, linear molecules, and non-linear molecules have orbitals which can be labeled either according to the symmetry appropriate for that isolated species or for the species in an environment which produces lower symmetry. These orbitals should be viewed as regions of space in which electrons can move, with, of course, at most two electrons (of opposite spin) in each orbital. Specification of a particular occupancy of the set of orbitals available to the system gives an electronic configuration. For example,... 

Recall that the symmetry labels e and o refer to the symmetries of the orbitals under reflection through the one Cy plane that is preserved throughout the proposed disrotatory closing. Low-energy configurations (assuming one is interested in the thermal or low-lying photochemically excited-state reactivity of this system) for the reactant molecule and their overall space and spin symmetry are as follows ... 

For the given orbital oeeupations (eonfigurations) of the following systems, determine all possible states (all possible allowed eombinations of spin and spaee states). There is no need to form the determinental wavefunetions simply label eaeh state with its proper term symbol. One method eommonly used is Harry Grays "box method" found in Eleetrons and Chemical Bonding. 

Now look at octahedral complexes, or those with any other environment possessing a centre of symmetry e.g. square-planar). These present a further problem. The process of violating the parity rule is no longer available, for orbitals of different parity do not mix under a Hamiltonian for a centrosymmetric molecule. Here the nuclear arrangement requires the labelling of d functions as g and of p functions as m in centrosymmetric complexes, d orbitals do not mix with p orbitals. And yet d-d transitions are observed in octahedral chromophores. We must turn to another mechanism. Actually this mechanism is operative for all chromophores, whether centrosymmetric or not. As we shall see, however, it is less effective than that described above and so wasn t mentioned there. For centrosymmetric systems it s the only game in town. 

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