Perfect-pairing orbitals

Figure 1. GVB-perfect pairing orbitals for the ground state of CO A and B show the orbitals of the 12-bonds for one of the CO double bonds C and D show the orbitals of the two lone pairs on one of the oxygen atoms.

The number No of occupied valence SCF orbitals in a molecule is typically less than the total number Nmb of orbitals in the minimal valence basis sets of all atoms. The full valence MCSCF wavefunction is the optimal expansion in terms of all configurations that can be generated from N b molecular orbitals. Closely related is the full MCSCF wavefunction of all configurations that can be generated from Ne orbitals, where Nc is the number of valence electrons, i.e. each occupied valence orbital has a correlating orbital, as first postulated by Boys (48) and also presumed in perfect pairing models (49,50), We shall call these two types of frill spaces FORS 1 and FORS 2. In both, the inner shell remains closed. 

One of the most useful types of constraint is the restriction of the spin coupling to just a single mode. Many molecular systems are described rather well by the perfect pairing mode of spin coupling, for example. A useful alternative, especially when this is not the case, is to base the structure coefficients on the CASSCF wavefunction in the VB orbital basis ... 

The spin-coupling pattern go for the reordered orbital set is largely dominated by its perfect-pairing component [ qq-s = in the Kotani basis, see Eq. (5)], in... 

No matter whether calculated within the perfect pairing VB approach or by the spin-coupled VB approach, in both cases the CC hybrid orbital extends outside the three-membered ring as expected by the schematic representations in Figure 7, but also inside... 

The product of bond wave functions in Equation 3.8, involves so-called perfect pairing, whereby we take the Lewis structure of the molecule, represent each bond by a HL bond, and finally express the full wave function as a product of all these pair-bond wave functions. As a rule, such a perfectly paired polyelectronic VB wave function having n bond pairs will be described by 2" determinants, displaying all the possible 2x2 spin permutations between the orbitals that are singlet coupled. 

This LBO-based wave function is not a VB wave function. Nevertheless, it represents a Lewis structure, and hence also a pictorial analogue of a perfectpairing VB wave function. The difference between the LBO and VB wave functions is that the latter involves electron correlation while the former does not. As such, in a perfectly paired VB wave function, based on CF orbitals, each localized Be—H bond would involve an optimized covalent—ionic combination as we demonstrated above for H2 and generalized for other 2e bonds. In contrast, the LBOs in Equation 3.65 possess some constrained combination of these components, with exaggeration of the bond ionicity. 

It follows from the above analysis that the rabbit-ears and canonical MO representations of the water s lone pairs are both perfectly correct, as they lead to equivalent wave functions for the ground state of water, as well as for its two ionized states. Both representations account for the two ionization potentials that are observed experimentally. This example illustrates the well-known fact that, while the polyelectronic wave function for a given state is unique, the orbitals from which it is constructed are not unique, and this holds true even in the MO framework within which a standard localization procedure generates the rabbit-ear lone pairs while leaving the total wave function unchanged. Thus, the question what are the true lone-pair orbitals of water is not very meaningful. 

The generalized valence bond (GVB) method was the earliest important generalization of the Coulson—Fischer idea to polyatomic molecules (13,14). The method uses OEOs that are free to delocalize over the whole molecule during orbital optimization. Despite its general formulation, the GVB method is usually used in its restricted form, referred to as GVB SOPP, which introduces two simplifications. The first one is the perfect-pairing (PP) approximation, in which only one VB structure is generated in the calculation. The wave function may then be expressed in the simple form of Equation 9.1, as a product of so-called geminal two-electron functions ... 

This strong orthogonality constraint, while seemingly a restriction, is usually not a serious one, since it applies to orbitals that are not expected to overlap significantly. On the other hand, the orbitals (

perfectly paired GVB wave function generated under the constraint of zero-overlap between the orbitals of different pairs. 

GVB Generalized valence bond. A theory that employs CF orbitals to calculate electronic structure with wave functions in which the electrons are formally coupled in a covalent manner. The simplest level of the theory is GVB PP (PP-perfect pairing), in which all the electrons are paired into bonds, as in the Lewis structure of the molecule. 

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... 

Among the VB related methods existent in the literature, besides GVB and SCVB, it is worth mentioning the VB-SCF and the BOVB (breathing orbital valence bond) methods [3]. The VB-SCF method incorporates orbital optimization to the classical VB scheme. When one has more than one important perfect pairing scheme (or resonance , but see the next Section) the BOVB method can be utilised. More recently McWeeny also presented his version of the classical VB method including orbital optimization and multistructural capabilities [20]. 

When the orbitals are ordered so that the first two are the inner orbitals and, if a valence orbital is even-numbered (odd-numbered), its symmetry-equivalent counterparts also are even-numbered (odd-numbered), then the spin part of the SC wavefunction is dominated by the perfect-pairing Yamanouchi-Kotani (YK) spin function, with a coefficient exceeding 0.99. The coefficients of the other 13 YK functions are all smaller than 0.01. 

---