Chemistry Reference
In-Depth Information
where
F
ij
= 〈 i | H | j 〉, and S
ij
= 〈 i | j 〉.
(4.6)
If i = j, then F
ij
is called the
Coulomb integral
and it represents the energy of an
electron from an atomic orbital. If i
≠
j, then F
ij
is called the
bonding integral
and
represents the energy of interaction of two atomic orbitals. The term S
ij
is called the
covering integral
and is a measure of the interpenetration of two atomic orbitals.
The presentation of the methods of solving Equations (4.4) and (4.5) and of calculat-
ing the values of the coefficients is too complex to cover in sufficient detail in this
chapter.
The term c
mi
2
is considered the charge induced by the orbital
i
in atom
m
. The
charge on a certain atom is also induced by orbitals of the neighboring atoms. These
orbitals might or might not participate in the bond between the two atoms (Roothaan
1951). The total charge q
m
of the atomic electronic cloud for atom
m
is calculated
with Formula (4.7).
q
m
= Σ(c
i
2
+ c
i
· c
j
· S
ij
)
(4.7)
The net charge (S) of atom
m
is given by the difference between the nuclear
charge Z and the charge, q
m
.
S = Z − q
m
(4.8)
The bond order (B) between any two atoms depends on the value of the coeffi-
cients in Equation (4.3) as calculated using Formula (4.9).
B = 2 · Σ c
mi
· c
mj
(4.9)
The bond order is calculated for any pair of atoms, but has high values only if the
atoms are neighbors. The smaller the value of B for two neighboring atoms, the more
ionic is the character of the chemical bond. On the other hand, the larger the value of
B, the more multiple is the character (aromatic, double, triple) of the chemical bond.
The calculation methods used by the quantum mechanics programs are parameter-
ized so that they calculate the bond order B ~ 1 for the single chemical bond, B ~ 1.5
for the aromatic chemical bond, B ~ 2 for the double chemical bond, and B ~ 3 for the
triple chemical bond.
Figure 4.4
presents the values of the bond orders calculated for the bonds in the
cycle of 6 atoms of the two isomers in the guanine molecule. In isomer A, the C-N
bond in the amide group is single, and the other bonds between the heavy atoms
(other than hydrogen) are calculated as aromatic, even the C = O or C-NH
2
bonds. In
the B isomer, all bonds between the heavy atoms are calculated as aromatic.
Knowledge of the bond order allows the grouping of that particular chemical
bond in a specific category (see Table 4.1), the counting of the chemical bonds, the
counting of the chemical bonds of a certain type, and the calculation of the Shannon
entropy of the chemical bonds. Furthermore, the value of the bond order allows the
identification of the bound and unbound atoms, after empirically setting a minimum
