Computational Chemistry

be more widely applicable than those from the Lewis picture, because electron pairs
are not localized between atoms in a clean pairwise manner; thus a weak bond, like
a hydrogen bond or a long single bond, might be expected to have a bond order of
less than one. However, there is no unique definition of bond order in computational
chemistry, because there seems to be no single, correct method to assign electrons
to particular atoms or pairs of atoms [ 47 ]. Various quantum mechanical definitions
of bond order can be devised [ 48 ], based on basis-set coefficients. Intuitively, these
coefficients for a pair of atoms should be relevant to calculating a bond order, since
the bigger the contribution two atoms make to the wavefunction (whose square is a
measure of the electron density;Section 4.2.6), the bigger should be the electron
density between them. In the SHM the order of a bond between two atoms Aiand Bj
is defined as

Bi;j¼ 1 þ

X

all occ

ncicj (4.70)

Here the 1 denotes the single bond of the ubiquitous spectatorsbond framework,
which is taken as always contributing asbond order of unity. The other term is the
pbond order; its value is obtained by summing over all the occupied MOs the
number of electronsnin each of these MOs times the product of thec’s of the two
atoms for each MO. This is illustrated in these examples:
Ethene.The occupied orbital isc 1 , which has 2 electrons), and the coefficients
ofc 1 andc 2 for this orbital are 0.707, 0.707 (Eq.4.66). Thus

Bi;j¼ 1 þ

X

all occ

ncicj¼ 1 þ 2 ð 0 : 707 Þ 0 : 707 ¼ 1 þ 1 : 000 ¼ 2 : 000

which is reasonable for a double bond. The order of thesbond is 1 and that of the
pbond is 1.
The ethene radical anion.The occupied orbitals arec 1 , which has 2 electrons,
andc 2 , which has 1 electron; the coefficients ofc 1 andc 2 forc 1 are 0.707, 0.707
and forc 2 , 0.707,"0.707 (Eq.4.66). Thus

Bi;j¼ 1 þ

X

all occ

ncicj¼ 1 þ 2 ð 0 : 707 Þ 0 : 707 þ 1 ð 0 : 707 Þð" 0 : 707 Þ

¼ 1 þ 1 " 0 : 500 ¼ 1 : 500

Thepbond order of 0.500 (1.500"sbond order) accords with two electrons in
the bonding MO and one electron in the antibonding orbital.

4.3.5.5 Atomic Charges

In an intuitive way, the charge on an atom might be thought to be a measure of the
extent to which the atom repels or attracts a charged probe near it, and to be

142 4 Introduction to Quantum Mechanics in Computational Chemistry