54
4 Representations
has determinant zero. This requirement is written as
D(C 3 ) − λI
= 0
(4.14)
Here, the vertical bars denote the determinant. Equation (4.14) is called the secular
equation. It has the form of a simple cubic equation in the eigenvalue λ:
−λ
3 + 1 = 0
(4.15)
This equation is the Euler equation. It has three roots:
λ m = exp
2mπi
3
(4.16)
where m can take the values −1, 0, +1. What we have just performed is a matrix
diagonalization of the representation matrix. We obtain at once not one but three
eigenvalues. The eigenfunction corresponding to a given root can now be found by
introducing this λ value in the system of equations, Eq. (4.13). Since the system
is homogeneous, the three unknown coefficients can be determined only up to a
constant factor. We find the absolute values of these vector coefficients by invoking a
normalization condition that requires the vectors to be of unit length. The simplified
normalization condition, neglecting overlap integrals, reads:
|c A |
2 +|c B |
2 +|c C |
2 = 1
(4.17)
In this way we obtain three SALCs, each characterized by a different eigenvalue for
the symmetry operator:
|ψ 0 =
1
√
3
|1s A +|1s B +|1s C
|ψ +1 =
1
√
3
|1s A ǫ|1s B +ǫ|1s C
|ψ −1 =
1
√
3
|1s A +ǫ|1s B +¯ ǫ|1s C
(4.18)
where ǫ = exp 2πi/3. The set of corresponding eigenvalues is called the spectrum of
the operator. It will be evident that this spectrum consists of the cube roots of 1, since
operating with ˆ
C 3 three times in succession is equivalent to applying the identity
operator:
ˆ
C
3
3 |ψ m =λ
3 |ψ m = ˆ
E|ψ m
(4.19)
For any operator, one can find eigenfunctions by simply diagonalizing the corresponding representation matrix. However, our objective is more ambitious. We want
to obtain functions that are not only adapted to a single symmetry element but to the
group as a whole. This really amounts to finding SALCs for the set of group generators, since adaptation to the generators implies that the function is adapted to any
4 Representations
has determinant zero. This requirement is written as
D(C 3 ) − λI
= 0
(4.14)
Here, the vertical bars denote the determinant. Equation (4.14) is called the secular
equation. It has the form of a simple cubic equation in the eigenvalue λ:
−λ
3 + 1 = 0
(4.15)
This equation is the Euler equation. It has three roots:
λ m = exp
2mπi
3
(4.16)
where m can take the values −1, 0, +1. What we have just performed is a matrix
diagonalization of the representation matrix. We obtain at once not one but three
eigenvalues. The eigenfunction corresponding to a given root can now be found by
introducing this λ value in the system of equations, Eq. (4.13). Since the system
is homogeneous, the three unknown coefficients can be determined only up to a
constant factor. We find the absolute values of these vector coefficients by invoking a
normalization condition that requires the vectors to be of unit length. The simplified
normalization condition, neglecting overlap integrals, reads:
|c A |
2 +|c B |
2 +|c C |
2 = 1
(4.17)
In this way we obtain three SALCs, each characterized by a different eigenvalue for
the symmetry operator:
|ψ 0 =
1
√
3
|1s A +|1s B +|1s C
|ψ +1 =
1
√
3
|1s A ǫ|1s B +ǫ|1s C
|ψ −1 =
1
√
3
|1s A +ǫ|1s B +¯ ǫ|1s C
(4.18)
where ǫ = exp 2πi/3. The set of corresponding eigenvalues is called the spectrum of
the operator. It will be evident that this spectrum consists of the cube roots of 1, since
operating with ˆ
C 3 three times in succession is equivalent to applying the identity
operator:
ˆ
C
3
3 |ψ m =λ
3 |ψ m = ˆ
E|ψ m
(4.19)
For any operator, one can find eigenfunctions by simply diagonalizing the corresponding representation matrix. However, our objective is more ambitious. We want
to obtain functions that are not only adapted to a single symmetry element but to the
group as a whole. This really amounts to finding SALCs for the set of group generators, since adaptation to the generators implies that the function is adapted to any