Computational Chemistry

Nr¼N 2 ¼nrþ

1

2

X

s 6 ¼r

nr=s¼n 2 þ

1

2

ðn 2 = 1 Þ¼ 1 : 2864 þ

1

2

ð 0 : 5114 Þ¼ 1 : 5421

The sum of all theNron He has only one term,N 2 , since there is only one basis
function on He:

NA¼NHe¼

X

r 2 He

Nr¼N 2 ¼ 1 : 5421

The charge on He,qHe, is the algebraic sum of the gross electronic population
and the nuclear charge:

qA¼qHe¼ZHe#NHe¼ 2 # 1 : 5421 ¼ 0 : 4579

The charges sum to 0.5423þ0.4579¼1.000, the total charge on the molecule.
The less positive charge on helium is in accord with the fact that electronegativity
increases from left to right along a row of the periodic table.
H–He bond order– For this we use Eq. (5.218);nr/sis summed for all overlaps
between basis functions on atoms A and B. There is only one such overlap, that
betweenf 1 andf 2 , so

bAB¼bHHe¼

X

r;s 2 A;B

nr=s¼n 1 = 2 ¼ 2 ð 0 : 2557 Þ¼ 0 : 5114

Note that the elements of the population matrix (PS) sum to the number of
electrons in the molecule: 0.2020þ1.2864þ0.2557þ0.2557¼2.000. This is
expected, since the diagonal elements are the number of electrons in the “atomic
space” of the basis functions, and the off-diagonal elements are the number of
electrons in the overlap space of the basis functions.
The Mulliken approach to population analysis has problems; for example, it
sometimes assigns more than two electrons, and sometimes a negative number of
electrons, to an orbital. It is also fairly basis-set dependent (Hehre, Radom, Schleyer
and Pople compare Mulliken charges for a variety of molecules using the STO-3G,
3–21G()and 6–31G basis sets: ref. [ 1 g], pp. 337–339). Partitioning half of the
electrons “arbitrarily” into the overlap region is not as serious as one might have
thought, because in a series of calculations a meaningful trend can emerge, even if a
charge or bond order taken in isolation is of dubious quantitative significance. Other
approaches to manipulating basis function coefficients for partitioning electrons
among orbitals and thus calculating charges and bond orders are those of Mayer
[ 258 ] and L€owdin [ 259 ] and the natural population analysis (NPA) of Weinhold
[ 260 ]. One point of interest in the Mayer method is that it seems to be the only one
that assigns the hydrogen molecule ion, Hþ 2 , with one electron, the intuitively
sensible bond order of 0.5, rather than 0.25 [ 261 ]. Mayer bond orders appear to have

5.5 Applications of the Ab initio Method 351