Structural Coefficients and Weights of Valence Bond Wave Functions

6 STRUCTURAL COEFFICIENTS AND WEIGHTS OF VALENCE BOND WAVE FUNCTIONS 

Once a wave function is available and is written as a linear combination of VB structures as in Equation 3.54, the major VB structures can be distinguished from the minor ones by consideration of the magnitudes of their respective coefficients. 

More generally, one has to consider the weights of VB structures, which are quantitatively related to physical properties like electron densities, net charges, and so on. According to the popular Chirgwin—Coulson formula (5), the weight of a given structure, d x, is defined as the square of the coefficient plus one-half of the overlap population terms with all the other structures  

This formula is the VB analogue of the Mulliken population in MO-based calculations. The VB weights sum to unity if the wave function 1P, in Equation 3.54, is normalized. However, Equation 3.55 can be used even if the VB structures are not normalized, or even if represents an AO-based determinant rather than a VB structure. In such a case, it is useful to note that with the definition of the weights as in the Chirgwin—Coulson formula, the weight of a VB structure is equal to the sum of the weights of its constituting VB determinants. 

Other definitions have also been proposed, such as the Lowdin weights (6), Equation 3.56, and the inverse weights (7), Equation 3.57. 

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