The Relationship between MO and VB Wave Functions

We now consider the difference between the MO and VB descriptions of an electronic system, at the simplest level of both theories. As we shall see, in the cases of one-electron, three-electron, and four-electron interactions between two centers, there is no real difference between the two theories, except for a matter of language. However, the two theories do differ in their description of the two-electron bond. Let us consider, once again, the example of H2, with its two AOs a and b, and examine first the VB description, dropping normalization factors for simplicity. 

Let us now turn to the MO description. Bringing together two hydrogen atoms leads to the formation of two MOs, a and a (bonding and antibonding, respectively) see Eq. [14]. 

At the simple MO level, the ground state of H2 is described by I mo, in which the bonding ct MO is doubly occupied. Expansion (see Appendix for details) of this MO determinant into its AO determinant constituents leads to Eq. [15]  

It follows that mixing the two configurations mo and with different coefficients as in Eq. [17] will lead to a wave function mo-ci in which the covalent and ionic components have 

Since Ci and Ci are variationally optimized, expansion of mo-ci should lead to exactly the same VB function as vB-fuii in Eq. [5], leading to the equalities expressed in Eq. [18b] and to the equivalence of mo-ci and vB-fuii- The equivalence also includes the Coulson-Fischer wave function (Eq [6]) which, as we have seen, is equivalent to the VB-full description. 

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