Spin coupled orbitals

Figure 4. Spin-Coupled orbitals for H2 (left) and H (right).

Figure 5. Spin-Coupled orbitals for one of the C-H bonds in CH4. The two Spin-Coupled orbitals showing their overlap (left), the orbital on carbon (middle) and...

In order to reduce further the Cl expansion, it is necessary to impose restrictions to the possible spin couplings within each group. The (3og,3au) pair, for example, contains three spin coupled orbital products ... 

Fig. 3. Spin-coupled orbital (fq in benzene — throughout this review we present contour plots of the square modulus of the orbitals,

It is important to bear in mind that each spin-coupled orbital contains just one electron. Thus, for example, if orbital cp2 is localized about nucleus A and cp2 is localized about nucleus B, then this is equivalent to saying that the electrons tend to keep out of each other s way their Coulombic repulsions are reduced, and so the state becomes considerably more stable. 

The spin-coupled wavefunction provides an improvement over the SCF energy of 199 kJ mol-1 (0.0758 hartree), which is considerable. This arises from the effects of electron correlation in the jr-electron system. The- distortion of the spin-coupled orbitals for benzene occurs because of the small amount of ionic character needed to describe the C — C n bonds. Ionic structures in spin-coupled theory are those in which one or more of the orbitals is allowed to be doubly occupied. Allowing for the different numbers of allowed spin functions, and including the spin-coupled configuration, a total of 175 VB structures can be generated in this way. A... 

Table 1. Overlap integrals between the spin-coupled orbitals for benzene...

We found that these more sophisticated spin-coupled calculations, which used larger basis sets with polarization functions on all of the atoms and which allowed the a orbitals to relax, produced a picture of bonding in the 7t-electron system of benzene which is practically identical to that described earlier. As before, we found six equivalent spin-coupled orbitals which are transformed into one another by successive C6 rotations. The overlaps between the orbitals, ordered cpa to cp6 around the ring, are reported in Table 1. In this case, the electron correlation effects incorporated in the spin-coupled model provide an energy improvement over the SCF description of 170 kJ mol - with a further lowering of 20 kJ mol -1 on including spin-coupled ionic structures. 

We found that the replacement of one C — H unit in benzene by a nitrogen atom resulted in only small changes in the form of the spin-coupled orbitals, in the relative importance of the different modes of spin coupling, and in our estimate of the resonance... 

Fig. 4. Spin-coupled orbitals for pyrimidine. The view corresponds to that in Fig. 3(b), and the molecule is orientated as in Fig. 1...

The spin-coupled description of the rt-electron systems in the heteroaromatic molecules furan, pyrrole, thiophen and thiazole is also based on localized but non-orthogonal orbitals. The spin-coupled orbitals for pyrrole are shown in Fig. 5. There is one orbital localized on each carbon atom and one on the heteroatom ([Pg.50]

One of the great strengths of MO theory is the guidance it provides for the assignment of photoelectron spectra by means of Koopmans theorem. Using physical assumptions which closely parallel those of Koopmans theorem, we have used the spin-coupled orbitals to examine simple valence bond estimates of the ionization potentials. We find that the results for the lowest ionization potentials are at least as good as those derived from Koopmans theorem, while the higher potentials appear to be considerably more reliable. 

Fig. 6. Spin-coupled orbitals cpt and cp2 in borazine. Two other orbitals are related by symmetry to each of these. Top row at one of the N atoms in a cr mirror plane. Bottom row in the plane 1 bohr above the ah molecular plane...

Figure 1. Symmetry-unique spin-coupled orbitals for the Diels-Alder reaction.

For this system, we find that the spin-coupled orbitals do not remain associated with the same first-row atom throughout the reaction. Instead, orbital /2 from the ethyne moiety becomes a linear combination of an sp -like hybrid from the ethyne and another such hybrid from the HCNO, as is shown for the transition state in the middle column of Figure 5. After the transition state, this orbital becomes almost entirely associated with the HCNO carbon atom. 

Figure 4. Various representations of spin-coupled orbital <)>, for benzene. Left contours in the horizontal plane 1 bohr above the molecular plane. Centre contours in a vertical mirror plane. Right a representative isosurface (3-D contour).

In the subsequent spin-coupled calculations [9], the < > were optimized as linear combinations of all the LMOs which correspond to X—Y bonds, plus all the virtual orbitals. This scheme is entirely equivalent to expanding the in the full basis atomic basis set, except that it maintains the orthogonality between the active orbitals and the inactive space, which consists of all the other doubly-occupied MOs. The spin function 0 was fully optimized in the full spin space. The active orbitals were thus fully optimized without constraints on their form, on the degree of localization, on the overlaps between them, or on the mode of coupling the electron spins. Nevertheless, we found for each molecule that the optimized spin-coupled orbitals consist of pairs, each clearly associated with a particular two-centre bond, and with predominantly singlet coupling of the electron spins. For example, we show in Figure 2 the pair of spin-coupled... 

Calculations of this type have also been performed for PXn (n=3,5) and SXn (n=2,4) fluorides and chlorides [9]. Much the same basic picture emerges for all of these systems, whether normal octet or hypercoordinate, with the variations in the amount of central-atom character in the two-centre spin-coupled orbital reflecting the polarity of the particular bond. Analogous descriptions were found to apply for XeFn (n=2,4) and for SiX ions (X=H,F). 

Figure 3a Spin-coupled orbitals for the out-of-plane n system of CH2N2.

The four spin-coupled orbitals for the out-of-plane 7i-electron system of diazomethane are illustrated in Figure 3a as contours in the a mirror plane, perpendicular to the molecular plane. Each of these orbitals (7ij-7i4) takes the form of a deformed 2pK function, slightly distorted towards one of the neighbouring centres, with one orbital on each of the terminal heavy atoms and two on the... 

Figure 4. Spin-coupled orbitals for SOF2, as described in the text... 

The molecules PF5 jCH,), (ra=l-3) all adopt trigonal bipyramidal structures, with a preference for the methyl substituents to occupy equatorial positions. We have found [29] that the spin-coupled description of the P—F bonds in these systems resemble closely that for the parent molecule, PF5. The two spin-coupled orbitals that describe a typical P—CH3 bond are shown in Figure 5. The closer match of electronegativities results in a phosphorus-based spx-like hybrid that exhibits less delocalization onto the methyl group, and in some delocalization of the methyl-based sp -like hybrid back onto phosphorus. These bonds are clearly much less polar than the P—F bonds, as we would have anticipated. 

Figure 5 Spin-coupled orbitals for a typical P—CH3 bond in fluorophosphoranes.

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