Non-paired spatial orbital structure

After the CASSCF calculation with the above choice of orbitals, in order to perform an efficient VB analysis, it is better in this case to resort to an overcomplete non-orthogonal hybrid set. The five active orbitals, in fact, can be split into ten hybrids, in term of which the VB transcription of the wavefunction turns out to be the simplest and the most compact. Such kinds of overcomplete basis sets are commonly used in constructing the so called non-paired spatial orbital structures (NPSO, see for example [35]), but it should be remarked that their use is restricted to gradient methods of wavefunction optimization, such as steepest descent, because other methods, which need to invert the hessian matrix (like Newton--Raphson) clearly have problems with singularities. 

THE ONE-ELECTRON BOND AND NON-PAIRED SPATIAL ORBITAL STRUCTURES... 

It may be noted that instead of invoking resonance between the standard Lewis stmctures of Figure 2-3, it is also possible to use the Linnett non-paired spatial orbital structures ° displayed in Figure 2-4 for some of them. 

Figure 2-4 Linnett non-paired spatial orbital structures, with pseudo one-electron bonds.

Possibly the most convenient valence-bond structures for I, and XeFj are the Linnett non-paired spatial orbital structures (8) and (9), each of which two ( pseudo " ) one-electron a bonds. 

In Chapter 23, comparisons are made between the wave-functions for Lewis, increased-valence and non-paired spatial orbital structures, and the Rundle-Pimentel molecular orbital formulation. 

We shall now examine in more detail some wave-functions for increased-valence structures, and compare them with wave-functions that may be constructed for standard Lewis and Linnett non-paired spatial orbital structures, as well as with the delocalized molecular orbital wave-functions. 

As discussed in Section 23-4, resonance between the Lewis structures (l)-(4), with the same atomic orbitals used in each Lewis structure, is equivalent to using the non-paired spatial orbital structure (10). The procedure described in Ref 22 can be used to construct increased-valence or non-paired spatial orbital wavefimctions when different atomic orbitals are used in different Lewis structures. 

Looking at the composition of these orbitals, the VB structure (22) is very similar to that involved in the Linnett-type non-paired spatial orbital representation for a four electron, three-centre bonding unit [23], discussed by Harcourt in Ref. [24]. The only difference is related to the number of hybrids involved in the Harcourt paper only three hybrids, one for each atom, are considered while here, owing to the use of an extended basis set, the four localized orbitals in Eq. (22) derive from the six hybrids described above. 

Looking at the shape of these orbitals, the VB structure (15) is very similar to that involved in the Linnett-type non-paired spatial orbital representation for a four electron. 

The lengths of their I-I bonds are longer than the I-l single bond length of 2.67 A for free. We can account for these observations by inspection of the increased-valence structures (30), (32) and (34), which we may derive from the Lewis or non-paired spatial oibital stmctures (29), (31) and (33). For the latter three structures, we have subdivided the ions into + r + Ij, Ij +1, + Ij, and Ij + r + Ij + r +12 components, and used the non-paired spatial orbital stmctuie (8) for I3. All of the bonds are o-bonds. 

The geometry of the electronic configuration in the pseudohalide ions can be treated in terms of Linnett s non-paired spin orbital approach (84). Six possible resonance structures can be drawn up as shown below (84). The lines indicate a pair of electrons of opposite spin in the same spatial orbital, the dotted lines indicate a pair of different spatial orbitals while o and x represent electrons of different spin. 

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