Semilocalized AOs

Valence Bond Wave Functions with Semilocalized AOs... 

CF orbitals Coulson-Fisher-type orbitals. These are semilocalized AOs, also called overlap enhanced orbitals (OEOs) which are localized on a given center, but have small delocalization tails on other centers. Special cases of CF AOs are BDOs (see above). 

Here, a and b are purely localized AOs, while Coulson—Fischer orbitals energy minimization, are generally not very delocalized (e < 1), and as such they can be viewed as distorted orbitals that remain atomic-like in nature. However, minor as this may look, the slight delocalization renders the Coulson—Fischer wave function equivalent to the VB-full wave function (Eq. 3.4a) with the three classical structures. A straightforward expansion of the Coulson—Fischer wave function leads to the linear combination of the classical structures in Equation 3.6. 

Note that if instead of using purely localized AOs for a and b, we use semilocalized Coulson-Fischer orbitals, Equation 3.37 will no more be the... 

All the VB methods that deal with semilocalized orbitals use a generalization of the Coulson—Fischer idea (12), whereby a bond is described as a singlet coupling between two electrons in nonorthogonal orbitals that possess small delocalization tails resulting from the variational orbital optimization. Albeit formally covalent, this description implicitly involves some optimal contributions of ionic terms, as a decomposition of the wave function in terms of pure AO determinants would show (see Eqs. 3.5 and 3.6). For a polyatomic... 

Note that the rules and formulas that are expressed above in the framework of qualitative VB theory are independent of the type of orbitals that are used in the VB determinants purely localized AOs, fragment orbitals or Coulson-Fischer semilocalized orbitals. Depending on the kind of orbitals that are chosen, the h and S integrals take different values, but the formulas remain the same. 

Here, p is the reduced resonance integral that we have just defined and S is the overlap between orbitals a and b. Note that if instead of using purely localized AOs for a and b, we use semilocalized Coulson-Fischer orbitals, Eq. [41] will not be the simple HL-bond energy but would represent the bonding energy of the real A B bond that includes its optimized covalent and ionic components. In this case, the origins of the energy would still correspond to the QC determinant with the localized orbitals. Unless otherwise specified, we will always use qualitative VB theory with this latter convention. 

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