4.5 Projection Operators
67
=
dim(Ω) 2
|G| 2
R,S
D
Ω
kl (R) ¯
D
Ω
kl (S)δ R,S
=
dim(Ω) 2
|G| 2
R
D
Ω
kl (R) ¯
D
Ω
kl ′ (R)
=
dim(Ω)
|G|
δ l,l ′
(4.52)
Hence, if the multiplicity is greater than one, an additional label preceding the irrep
label has to be introduced in order to distinguish SALCs with the same symmetry,
and, by varying the l index of the projector, all these can be projected out. Note
that the maximal invariance space of a symmetry group is bound to be the regular
representation; hence, multiplicities of an invariant function space cannot exceed
the dimensions of the irreps and thus will always be covered by the variation of
index l. If the multiplicity is smaller than dim(Ω), variation of l will give rise to
redundancies.
The action of the projector on an arbitrary function can be written as
ˆ
P
Γ
kl |f x =S
Γl
x
Φ
Γ
k
(4.53)
Hence, the projector takes out of the function an irreducible part that transforms as
the |Φ Γ
k SALC multiplied by an overlap factor, S Γl
x , which indicates the extent to
which the |Φ Γ
l SALC is present in this function. A very concise formulation of this
result can be achieved by the use of the Dirac notation. In this notation, the projector
is written as
ˆ
P
Γ
kl =
Φ
Γ
k
Φ
Γ
l
(4.54)
In the ket–bra combination, all the aspects of the projector come together. Let us
apply this to our function:
ˆ
P
Γ
kl |f x =
Φ
Γ
k
Φ
Γ
l
f x
=
Φ
Γ
k
Φ
Γ
l
f x
(4.55)
where the convention is followed that the juxtaposition of two vertical lines is contracted to one. Comparing Eqs. (4.54) and (4.55), one can identify the bracket:
S
Γl
x =
Φ
Γ
l
f x
(4.56)
When the projection operator acts (on the left) on a function |f x , it forms a bracket,
which is the overlap factor measuring how much of the |Φ Γ
l SALC is present in the
target. This is the “recognition” part of the projection. It then returns, as a result, the
desired SALC |Φ Γ
k multiplied by the overlap factor. This is the ladder aspect. As
an example, consider the action of the ˆ
P E projection operators on the |1s A orbital
67
=
dim(Ω) 2
|G| 2
R,S
D
Ω
kl (R) ¯
D
Ω
kl (S)δ R,S
=
dim(Ω) 2
|G| 2
R
D
Ω
kl (R) ¯
D
Ω
kl ′ (R)
=
dim(Ω)
|G|
δ l,l ′
(4.52)
Hence, if the multiplicity is greater than one, an additional label preceding the irrep
label has to be introduced in order to distinguish SALCs with the same symmetry,
and, by varying the l index of the projector, all these can be projected out. Note
that the maximal invariance space of a symmetry group is bound to be the regular
representation; hence, multiplicities of an invariant function space cannot exceed
the dimensions of the irreps and thus will always be covered by the variation of
index l. If the multiplicity is smaller than dim(Ω), variation of l will give rise to
redundancies.
The action of the projector on an arbitrary function can be written as
ˆ
P
Γ
kl |f x =S
Γl
x
Φ
Γ
k
(4.53)
Hence, the projector takes out of the function an irreducible part that transforms as
the |Φ Γ
k SALC multiplied by an overlap factor, S Γl
x , which indicates the extent to
which the |Φ Γ
l SALC is present in this function. A very concise formulation of this
result can be achieved by the use of the Dirac notation. In this notation, the projector
is written as
ˆ
P
Γ
kl =
Φ
Γ
k
Φ
Γ
l
(4.54)
In the ket–bra combination, all the aspects of the projector come together. Let us
apply this to our function:
ˆ
P
Γ
kl |f x =
Φ
Γ
k
Φ
Γ
l
f x
=
Φ
Γ
k
Φ
Γ
l
f x
(4.55)
where the convention is followed that the juxtaposition of two vertical lines is contracted to one. Comparing Eqs. (4.54) and (4.55), one can identify the bracket:
S
Γl
x =
Φ
Γ
l
f x
(4.56)
When the projection operator acts (on the left) on a function |f x , it forms a bracket,
which is the overlap factor measuring how much of the |Φ Γ
l SALC is present in the
target. This is the “recognition” part of the projection. It then returns, as a result, the
desired SALC |Φ Γ
k multiplied by the overlap factor. This is the ladder aspect. As
an example, consider the action of the ˆ
P E projection operators on the |1s A orbital