92
4 Representations
more generally the automorphism group of the molecular graph—acts transitively
on the set of atomic nodes. So, for any pair of atoms kl, there is a symmetry
element that will map k onto l, and then of course there is always an inverse
element that maps l onto k. The solution of the Hamiltonian matrix for such
a system can almost entirely be performed by symmetry arguments. The first step
consists in the construction of SALCs, using the projection operators on the atomic
orbital on site a:
|Φ
Ω
ik = ˆ
P
Ω
ik |φ A =
dim(Ω)
|G|
R∈G
¯
D
Ω
ik (R) ˆ
R|φ A
(4.140)
These functions are not yet normalized. This can be done later. Let us first consider
a general matrix element:
Φ
Ω
ik |H|Φ
Ω ′
jl
=
dim(Ω) 2
|G| 2
R,S
D
Ω
ik (R) ¯
D
Ω ′
jl (S) ˆ
Rφ A |H| ˆ
Sφ A
=
dim(Ω) 2
|G| 2
R,Q
D
Ω
ik (R) ¯
D
Ω ′
jl (RQ) ˆ
Rφ A |H| ˆ
RQφ A
=
dim(Ω) 2
|G| 2
R,Q
m
D
Ω
ik (R) ¯
D
Ω ′
jm (R) ¯
D
Ω ′
ml (Q)
φ A | ˆ
R
−1 H ˆ
R| ˆ
Qφ A
=
dim(Ω)
|G|
δ ΩΩ ′ δ ij
Q
¯
D
Ω
kl (Q)φ A |H| ˆ
Qφ A
(4.141)
The by-now experienced reader has recognized in the second line of this derivation
the use of a substitution, ˆ
S → ˆ
R ˆ
Q, as well as the invariance of the Hamiltonian
under the symmetry transformation in the third line. Let us now use this equation to
normalize the SALCs. This can be done by simply setting the Hamiltonian equal to
unity. Adopting the Hückel approximation, which neglects all overlaps between the
sites, we obtain:
Φ
Ω
ik |Φ
Ω
ik
=
dim(Ω)
|G|
Q
¯
D
Ω
kk (Q)δ Q,E =
dim(Ω)
|G|
(4.142)
Hence, the normalized SALCs should be redefined as
|Φ
Ω
ik =
dim(Ω)
|G|
R∈G
¯
D
Ω
ik (R) ˆ
R|φ A
(4.143)
The matrix elements are accordingly simplified to
Φ
Ω
ik |H|Φ
Ω
il
=
Q
¯
D
Ω
kl (Q)φ A |H| ˆ
Qφ A
(4.144)