Population Analysis and Bond Orders

A widely nsed (and widely criticized) method to analyze SCF MO wave functions is population analysis, introdnced by Mulliken. He proposed a method that apportions the electrons of an n-electron molecule into net populations in the basis functions Xr and overlap populations for all possible pairs of basis functions. 

For simplicity, we shall assume that the Cj, s and s s are real. The probability density associated with one electron in is 

Integrating this equation over three-dimensional space and using the fact that and the f s are normalized, we get 

By summing over the occupied MOs we obtain the MuUiken net population in Xr and the overlap population rirs for the pair Xr and Xs as 

The sum of all the net and overlap populations equals the total number of electrons n in the molecule (Prob. 15.23) = n. 

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