Essentials of Computational Chemistry

9.1 PROPERTIES RELATED TO CHARGE DISTRIBUTION 313
Mulliken partial atomic charge is then defined as
qk = Zk − Nk
(9.15)
where Z is the nuclear charge and Nk is computed according to Eq. (9.14).
With minimal or small split-valence basis sets, Mulliken charges tend to be reasonably
intuitive, certainly in sign if not necessarily in magnitude. Analysis of changes in charge as
a function of substitution or geometric change tends to be the best use to which Mulliken
charges may be put, and this can often provide chemically meaningful insight, as illustrated
in Figure 9.3.
The use of a non-orthogonal basis set in the Mulliken analysis, however, can lead to
some undesirable results. For instance, if one divides up the total number of electrons over
AO basis functions (in a fashion exactly analogous to that used for atoms), it is possible
for individual basis functions to have occupation numbers greater than 1 (which would be
greater than 2 in a restricted theory) or less than 0, and such a situation obviously can
have no physical meaning. In addition, the rule that all shared electrons should be divided
up equally between the atoms on which the sharing basis functions reside would seem to
ignore the possibly very different electronegativities of these atoms. Finally, Mulliken partial
charges prove to be very sensitive to basis-set size, so that comparisons of partial charges
Mulliken charge (a.u.)
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.1
H
O
H
H
H
O
H
C O
H
H
C O
H
H
0 20 40 60 80 100
Dihedral angle (°)
H
H
H
O
120 140 160 180
H
C O
Figure 9.3 AM1 Mulliken charges of the hydroxyl (circles) and oxonium (squares) oxygen atoms in
protonated dihydroxymethane as a function of HOCO + dihedral angle. Standard precepts of conformational
analysis suggest that hyperconjugation of hydroxyl oxygen lone-pair density (acting as a
donor) into the C–O + σ ∗ orbital (acting as an acceptor) may occur, and the effect is expected to be
maximal at a dihedral angle of 90 ◦ , and minimal at 0 ◦ and 180 ◦ . The computed Mulliken charges
on the oxygen atoms support hyperconjugation being operative, with about one-tenth of a positive
charge being transferred from the oxonium oxygen to the hydroxyl oxygen at a dihedral angle of 90 ◦
compared to 180 ◦ (an interpretation also consistent with geometric and energetic analysis, see Cramer
1992)
H
H