Computational Chemistry

measurable from the energy it takes to bring a probe charge from infinity up to near
the atom. However, this would tell us the charge at a point outside the atom, for
example a point on the van der Waals surface of the molecule, and the repulsive or
attractive forces on the probe charge would be due to the molecule as a whole.
Although atomic charges in molecules are generally considered to be experimen-
tally unmeasurable (but see Chapter 5, section 5.5.4,Charges and bond orders),
chemists find the concept very useful (thus calculated charges are used to parame-
terize molecular mechanics force fields –Chapter 3), and much effort has gone into
designing various definitions of atomic charge [ 48 , 49 ]. Intuitively, the charge on an
atom should be related to the basis set coefficients of the atom, since the more the
atom contributes to a multicenter wavefunction (one with contributions from basis
functions on several atoms), the more it might be expected to lose electronic charge
by delocalization into the rest of the molecule (cf. the discussion of bond order
above). In the SHM the charge on an atom Aiis defined as (cf. Eq.4.70)

qi¼ 1 "

X

all occ

nc^2 i (4.71)

The summation term is the charge density, and is a measure of the electronic
charge on the molecule due to thepelectrons. For example, having nopelectrons
(an emptyporbital, formally a cationic carbon) would mean apelectron charge
density of zero; subtracting this from unity gives a charge on the atom ofþ1. Again,
having twopelectrons in aporbital would mean apelectron charge density of 2 on
the atom; subtracting this from unity gives a charge on the atom of"1 (a filled
porbital, formally an anionic carbon). The application of Eq.4.71will be illustrated
using methylenecyclopropene (Fig.4.24).

4.3.5.6 Methylenecyclopropene

q 1 ¼ 1 "

X

all occ

nc^21 ¼ 1 " 2 ð 0 : 282 Þ^2 þ 2 ð 0 : 815 Þ^2

hi

¼ 1 " 1 : 487 ¼" 0 : 487

q 2 ¼ 1 "

X

all occ

nc^22 ¼ 1 " 2 ð 0 : 612 Þ^2 þ 2 ð 0 : 254 Þ^2

hi

¼ 1 " 0 : 878 ¼ 0 : 122

q 3 ¼q 4 ¼ 1 "

X

all occ

nc^23 ¼ 1 " 2 ð 0 : 523 Þ^2 þ 2 ð" 0 : 368 Þ^2

hi

¼ 1 " 0 : 817 ¼ 0 : 182

The results of this charge calculation are summarized in Fig.4.24; the negative
charge on the exocyclic carbon and the positive charges on the ring carbons are in
accord with the resonance picture (Fig.4.24), which invokes a contribution from
the aromatic cyclopropenyl cation [ 50 ]. Note that the charges sum to (essentially)
zero, as they must for a neutral molecule (the hydrogens, which actually also carry

4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 143