The Perfect Pairing Function

As an example, consider methane. If the carbon atom L-shell orbitals are arranged as tetrahedral hybrids, we can take the tat, t td configuration and combine this with an 3aSbScSd configuration of the four hydrogen atoms. Table 1 shows some numbers of states associated with these orbitals. It is 

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The apparently strange behaviour of qjy, namely the presence of a minimum at the CC1 distance of 2.1 A, could be a sign of resonance of structures involving diffuse orbitals on nitrogen, with the perfect pairing function (22) being representative of a more complex CASSCF wavefunction, or it could be simply related to the quality of the basis set. As anticipated, this is a preliminary result of work in progress a full account will be published elsewhere in due course. 

The choice of a single function from either set (36) or (37) does not permit such a useful physical interpretation, and may indeed lead to difficulties as the internuclear distance is varied. Thus if one chooses just the perfectly paired function from the set (36), as -R-> 00 one finds each N atom is described by a curious non-stationary state - the so-called valence state of the atom, about which there has been so much discussion in the literature.18 The choice of the set of functions (36) in which orbitals participating in a bond are directly coupled to each other is just the VB theory as proposed by Slater and Pauling,19 whereas the set (37) formed from atoms in specific L-S coupled states corresponds to the spin-valence theory employed by Heitler.20... 

It has so far been usual to consider almost exclusively the perfectly paired function (98), in which the spins of all the pairs y/(/, are coupled to form... 

Raimondi, Campion, and Karplus did an ab initio all-electron VB calculation on CH4 using a minimal AO basis set of STOs [M. Raimondi, W. Campion, and M. Karplus, Mol. Phys., 34,1483 (1977)]. Their wave function is a linear combination of 104 symmetry functions (and is a linear combination of a much larger number of individual functions) and contained 4900 Slater determinants. Very surprisingly, the function with the largest coefficient in the wave function is not the perfect-pairing function... 

Spin functions oo i and qo s are identical and define the perfect-pairing mode for a six-electron-singlet. 

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 ... 

The perfect pairing wave function and the valence state of carbon... 

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. 

The first seven configurations remain almost exclusively perfect-paired. As for the eighth configuration, it too turns out to be almost exclusively perfect-paired the coefficient of the perfectly-paired YK spin function is in fact 0.982. In any case, the fact that its first two orbitals are identical rules out nine YK spin functions out of fourteen, and symmetry requirements further mandate three linear constraints on the coefficients of the five allowed spin functions, so that only two of them are truly independent. Anyway, one can legitimately conclude that the seven-configuration wavefunction is qualitatively robust with respect to the inclusion of this kind of inner shell correlation. 

In essence, then, the general linear combination of couplings in the function (36) allows one to describe the dissociation process as a smooth recoupling of the orbitals from the perfectly paired state to the atomic coupling (2p3, iS) on each atom. 

In the spin-coupled description of a molecule such as SF6, the sulfur atom contributes six equivalent, nonorthogonal sp -like hybrids which delocalize onto the fluorine atoms. Each of these two-centre orbitals overlaps with a distorted F(2p) function and the perfect-pairing spin function dominates. Of course, using only 3s, 3px, 3p and 3pz atomic orbitals, we can at most form four linearly independent hybrid orbitals localized on sulfur, with a maximum occupancy of 8 electrons, as in the octet rule. However, the six sulfur+fluorine hybrids which emerge in the spin-coupled description are not linearly dependent, precisely because each of them contains a significant amount of F(2p) character. It is thus clear that the polar nature of the bonding is crucial. 

The perfect-pairing (PP) orbitals of this wave function clearly show the "lone-pairs" and "bond pairs" which are part of the language of the experimental chemist. This is in contrast to the molecular orbital description or to the GVB description with (7-7T restrictions where the lone pairs and "7T" bonds are not discernable from contour plots of the orbitals (2 ). It is somewhat reassuring that the wave function which gives the lowest variational energy (that of Figures 1 and 2a) also most closely coincides with the experimental chemist s traditional view of the bonding ( 3) ... 

A less trivial example is H2O. Although the H2O ionic VB structures D and E are very important, it is traditional to ignore the ionic structures and find the oxygen valence state in H2O from the perfect-pairing covalent function (15.149) corresponding to structure A. Removal of the H atoms from (15.149) gives... 

In the perfect pairing (PP) model, each pair of electrons is described by its own geminal. In contrast, the AGP and Pfaffian functions share the same geminal for all electron pairs. The perfect pairing geminal has a more constrained form than the previously described wave functions. In each geminal, a pair of active occupied orbitals, ifiir i, has a corresponding pair of active virtual orbitals, tA, ... 

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