Computational Chemistry

“electron density sum” in Eq. (5.210). Now if there arenielectrons in MOci, then
the contributions ofcito the electron population of basis functionfrand of the
overlap region betweenfrandfsare

nr;i¼nic^2 ri ð 5 : 211 Þ

and

nr=s;i¼nið 2 cricsiSrsÞð 5 : 212 Þ

The total contributions from all the MOs to the electron population infrand in
the overlap region betweenfrandfsare

nr¼

X

i

nr;i¼

X

i

nic^2 ri ð 5 : 213 Þ

and

nr=s¼

X

i

nr=s;i¼

X

i

nið 2 cricsiSrsÞð 5 : 214 Þ

The sums are over the occupied MOs, sinceni¼0 for the virtual MOs. The
numbernris theMulliken net populationin the basis functionfr, and the number
nr/sis theMulliken overlap populationfor the pair of basis functionsfrandfs. The
net population summed over allrplus the overlap population summed over allr/s
pairs equals the total number of electrons in the molecule.
The quantitiesnrandnr/sare used to calculate atom charges and bond orders.
TheMulliken gross population in the basis functionfris defined as the Mulliken
net populationnr(Eq. (5.211)) plus one half of all those Mulliken overlap popula-
tionsnr/s(Eq. (5.212)) which involvefr(of course for somefs,nr/smay be
negligible; e.g. for well-separated atomsSrsis very small):

Nr¼nrþ

1

2

X

s 6 ¼r

nr=s ð 5 : 215 Þ

The gross populationNris an attempt to represent the total electron population in
the basis functionfr; this is considered here to be the net populationnr, the
population that all the occupied MOs contribute tofrthrough the representation
offrin eachciby its coefficientcri(Eq. (5.213), plus one-half of the all the
populations in the overlap regions involvingfr(Fig.5.39b). Assigning tofrone-
half, rather than some other fraction, of the electron population in an overlap region
withfsis said to be arbitrary. Of course it is not arbitrary, in the sense that Mulliken
thought about it carefully and decided that one-half was at least as good as any other
fraction. One might imagine a more elaborate partitioning in which the fraction

5.5 Applications of the Ab initio Method 347