Chemistry Reference
In-Depth Information
By defining:
at
i
H
1e
at
j
H
ij
=
ψ
|
|
ψ
,
(1.17a)
at
i
at
j
S
ij
=
ψ
|
ψ
,
(1.17b)
the secular Eq. (1.16) is simplified to the expression:
N
at
c
j
H
ij
−
ES
ij
=
0
.
(1.18)
j
=
1
Therefore, with the LCAO approximation, Eq. (1.8) transforms to a system of
N
at
equations with
N
at
unknown parameters
c
j
. The resolution of Eq. (1.18) implies
that the determinant of the
H
ij
−
ES
ij
matrix has to be zero, otherwise we would
obtain the trivial solution
c
j
=
0
∀
j
, which obviously has no physical meaning.
Therefore,
H
11
−
ES
11
H
12
−
ES
12
...
H
1
N
at
−
ES
1
N
at
H
21
−
ES
21
H
22
−
ES
22
...
H
2
N
at
−
ES
2
N
at
=
.
0
(1.19)
.
.
.
.
H
N
at
1
−
ES
N
at
1
H
N
at
2
−
ES
N
at
2
...
H
N
at
N
at
−
ES
N
at
N
at
In conclusion, the energies
E
that satisfy Eq. (1.19) are associated to molecular
electronic states. Since Eq. (1.19) is an equation of
N
at
order, we obtain
N
at
energy
values
E
l
(
l
,
N
at
), that is, as many molecular levels as atomic orbitals. In
the simple example of H
2
discussed in Section 1.1,
N
at
=
=
1,
...
2 and both 1
s
atomic
orbitals combine to form bonding
u
MOs. In the case of N
2
(see Fig. 1.1), neglecting 1
s
core electrons, the combination of two
sp
and one
p
z
atomic orbitals per N atom leads to six MOs.
The easiest way to calculate the terms
H
ij
and
S
ij
is within the simple Huckel
approximation, where it is assumed that:
σ
g
and antibonding
σ
H
ii
=
(
E
α
)
i
,
(1.20a)
H
ij
=
(
E
β
)
ij
j
=
i
±
1
,
(1.20b)
S
ij
=
δ
ij
,
(1.20c)
where
δ
ij
stands for the Kronecker delta function (
δ
=
1 for
i
=
j
and
δ
=
0 for
i
0 unless the
i
th and
j
th orbitals are on adjacent
atoms (nearest neighbours). The term (
E
α
)
i
corresponds to the energy of the atomic
orbital
=
j
). Note that for
i
=
jH
ij
=
at
i
and (
E
β
)
ij
represents the nearest-neighbour resonance integrals. This
extremely simplified method works surprisingly well for small systems and has
the advantage that it permits the resolution of the secular equation by hand and
|
ψ
