Computational Chemistry

depends on the electronegativity difference between the atoms on whichfrand
fsreside, with the more electronegative atom getting the larger share of the electron
population. To get the charge on an atom A we calculate thegross atomic popula-
tion for A:

NA¼

X

r 2 A

Nr ð 5 : 216 Þ

This is the sum over all the basis functionsfron atom A (r∈A qualifying
the summation means “r belonging to A”) of the gross populations in eachfr
(Eq. (5.215); it involves all the basis functions on A and all the overlap regions
these functions have with other basis functionsfs. We can regardNAas the total
electron population on atom A (within the limits of the Mulliken treatment). The
Mulliken chargeon atom A, thenet chargeon A, is then simply the algebraic sum
of the charges due to the electrons and the nucleus:

qA¼ZA#NA ð 5 : 217 Þ

TheMulliken bond orderfor the bond between atoms A and B is the total
population for the A/B overlap region:

bAB¼

X

r;s 2 A;B

nr=s ð 5 : 218 Þ

The overlap population for basis functionsfrandfs(Eq. (5.214)) is summed
over all the overlaps between basis functions on atoms A and B.
Since the formulas for calculating Mulliken charges and bond orders
(Eqs. (5.211)–(5.218)) involve summing basis function coefficients and overlap
integrals, it is not too surprising that they can be expressed neatly in terms of the
density matrix (Section 5.2.3.6.4)Pand the overlap matrixS(Section 4.3.3). The
elements of the density matrixPare (cf. Eqs.5.208¼5.81)

Prs¼ 2

Xn

i¼ 1

cricsi ð 5 : 219 Þ

The matrix elementPrsis summed over all filled MOs (fromc 1 tocnfor the
ground electronic state of a 2n-electron closed-shell molecule); an example of the
calculation ofPwas given in Section 5.2.3.6.5. The elements of the overlap matrix
Sare simply the overlap integrals:

Srs¼

Z

frfsdv ð 5 : 220 Þ

From Eq. (5.219) it follows that the matrix (PS) obtained by multiplying
corresponding elements ofPandS,

348 5 Ab initio Calculations