Other Parameters Generated Through HMO Theory

Even though HMO theory cannot be relied upon for quantitatively correct predictions for some physical properties, it nevertheless provides a convenient framework for the development of a number of useful concepts in molecular bonding and reactivity. Among these are k electron density, charge density, bond order, and free valence. We calculate the electron density (p,) at each atom by summing the electron density at that atom for each occupied molecular orbital. We have defined ipi in equation 4.1  

the wave functions must be normalized. Solving equation 4.26 and taking the cross terms equal to 0 gives 

Since the integrals in square brackets are imity (see equation 4.10), the sum of the squares of the coefficients for each MO must equal 1. That is. 

Therefore, the square of the coefficient for in each wave function indicates the fraction of electron density to be found at the ith carbon atom when there is one electron in that i/ . If there are two electrons in a particular MO, then the electron density from that MO at the ith carbon atom is 2 c. To calculate the total electron density at a particular carbon atom, we must sum the electron density at that position in each of the MOs. The general expression for electron density at the ith position is then 

4 APPLICATIONS OF MOLECULAR ORBITAL THEORY AND VALENCE BOND THEORY 

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