4.1 Symmetry-Adapted Linear Combinations of Hydrogen Orbitals in Ammonia
53
While the function space is clearly invariant, i.e., it transforms into itself under the
rotation, the individual components are not: they are mutually permuted. Our objective is to find symmetry-adapted linear combinations (SALCs) that are invariant
under the operator, except, possibly, for a phase factor. The combinations we are
looking for are thus nothing other than eigenfunctions of the symmetry operator,
in the same way as solutions of the Schrödinger equation are eigenfunctions of the
Hamiltonian. Unlike the Hamiltonian eigenfunctions, however, which will usually
consist of linear combinations in an infinite Hilbert space, the present exercise is carried out in a space of three functions only since this space is already closed under
the operator. We shall solve this symmetry eigenvalue problem in a purely algebraic
way. Let |ψ m beaSALC:
|ψ m =
X=A,B,C
c X |1s X
(4.8)
which we shall again write as the product of a row vector and a column vector:
|ψ m =
|1s A | 1s B | 1s C
⎛
⎝
c A
c B
c C
⎞
⎠
(4.9)
The transformation of this function is then given by
ˆ
C 3 |ψ m =|f
⎛
⎝
001
100
010
⎞
⎠
⎛
⎝
c A
c B
c C
⎞
⎠
(4.10)
We now require this function to be an eigenfunction of the threefold rotation operator with eigenvalue λ:
ˆ
C 3 |ψ m =λ|ψ m
(4.11)
Combining Eqs. (4.10) and (4.11), we see that the function is an eigenfunction if
the product of the D matrix with the column vector of the coefficients returns the
coefficient column, multiplied by the eigenvalue λ, i.e.,
⎛
⎝
001
100
010
⎞
⎠
⎛
⎝
c A
c B
c C
⎞
⎠ = λ
⎛
⎝
c A
c B
c C
⎞
⎠
(4.12)
This equation can also be rewritten as
⎛
⎝
−λ 01
1 −λ 0
01 −λ
⎞
⎠
⎛
⎝
c A
c B
c C
⎞
⎠ = 0
(4.13)
Equation (4.13) forms a homogeneous system of equations in the three unknowns. It
will have solutions only if the matrix preceding the column vector of the unknowns
53
While the function space is clearly invariant, i.e., it transforms into itself under the
rotation, the individual components are not: they are mutually permuted. Our objective is to find symmetry-adapted linear combinations (SALCs) that are invariant
under the operator, except, possibly, for a phase factor. The combinations we are
looking for are thus nothing other than eigenfunctions of the symmetry operator,
in the same way as solutions of the Schrödinger equation are eigenfunctions of the
Hamiltonian. Unlike the Hamiltonian eigenfunctions, however, which will usually
consist of linear combinations in an infinite Hilbert space, the present exercise is carried out in a space of three functions only since this space is already closed under
the operator. We shall solve this symmetry eigenvalue problem in a purely algebraic
way. Let |ψ m beaSALC:
|ψ m =
X=A,B,C
c X |1s X
(4.8)
which we shall again write as the product of a row vector and a column vector:
|ψ m =
|1s A | 1s B | 1s C
⎛
⎝
c A
c B
c C
⎞
⎠
(4.9)
The transformation of this function is then given by
ˆ
C 3 |ψ m =|f
⎛
⎝
001
100
010
⎞
⎠
⎛
⎝
c A
c B
c C
⎞
⎠
(4.10)
We now require this function to be an eigenfunction of the threefold rotation operator with eigenvalue λ:
ˆ
C 3 |ψ m =λ|ψ m
(4.11)
Combining Eqs. (4.10) and (4.11), we see that the function is an eigenfunction if
the product of the D matrix with the column vector of the coefficients returns the
coefficient column, multiplied by the eigenvalue λ, i.e.,
⎛
⎝
001
100
010
⎞
⎠
⎛
⎝
c A
c B
c C
⎞
⎠ = λ
⎛
⎝
c A
c B
c C
⎞
⎠
(4.12)
This equation can also be rewritten as
⎛
⎝
−λ 01
1 −λ 0
01 −λ
⎞
⎠
⎛
⎝
c A
c B
c C
⎞
⎠ = 0
(4.13)
Equation (4.13) forms a homogeneous system of equations in the three unknowns. It
will have solutions only if the matrix preceding the column vector of the unknowns