10
1 Basic Concepts
These expressions produce projections that are not necessarily normalized to one, but
this can easily be done at the end of the process. The procedure is most conveniently
illustrated by deriving the singlet and triplet open-shell spin eigenfunctions with
M S = 0 for a two-electron in two-orbitals case. In the notation of Eq. 1.15 the two
determinants are
Φ 1 =|ϕ 1 ϕ 1 ϕ 2 ϕ 2 ...ϕ a ϕ b |=|ab|
Φ 2 =|ϕ 1 ϕ 1 ϕ 2 ϕ 2 ...ϕ a ϕ b |=|ab|
(1.31)
There are two possible spin eigenfunctions, singlet and triplet, with S equal to 0 and
1, respectively. The projection operators are directly obtained from Eq. 1.30
ˆ
P 0 = ˆ
S
2 − 2
ˆ
P 1 = ˆ
S
2 − 0
(1.32)
The result of applying ˆ
S 2 on Φ 1 is given in Eq. 1.27, and hence, the projection
operators give
ˆ
P 0 |ab|=( ˆ
S
2 − 2)|ab|=|ab|+|ab|−2|ab|=|ab|−|ab|
(1.33a)
ˆ
P 1 |ab|=( ˆ
S
2 − 0)|ab|=|ab|+|ab|
(1.33b)
The functions have to be multiplied by
1
√
2
to obtain the properly normalized expressions.
1.6 (a) Find the other two components of the triplet spin eigenfunctions by
applying the ladder operators on the M S = 0 component of the triplet function.
(b) Derive the singlet and triplet spin eigenfunctions by projection using Φ 2
of Eq. 1.31.
1.2.2 Spin Functions by Diagonalization
One way to find the eigenvalues and eigenvectors of an operator is to diagonalize
the matrix representation of the operator in a complete basis. Therefore, a natural
alternative to the projection method is the process of diagonalizing the matrix representation of the ˆ
S 2 operator. The basis of the matrix representation is formed by
the individual determinants. The resulting eigenvectors are the spin eigenfunctions
(linear combinations of these basis functions, the determinants) and the corresponding eigenvalues indicate the spin of the eigenfunction. The method is straightforward
in its application but can require a substantial amount of analytical work since all
matrix elements of ˆ
S 2 are needed, which can become rather cumbersome for systems
with an elevated number of unpaired electrons. The method is illustrated for a system
1 Basic Concepts
These expressions produce projections that are not necessarily normalized to one, but
this can easily be done at the end of the process. The procedure is most conveniently
illustrated by deriving the singlet and triplet open-shell spin eigenfunctions with
M S = 0 for a two-electron in two-orbitals case. In the notation of Eq. 1.15 the two
determinants are
Φ 1 =|ϕ 1 ϕ 1 ϕ 2 ϕ 2 ...ϕ a ϕ b |=|ab|
Φ 2 =|ϕ 1 ϕ 1 ϕ 2 ϕ 2 ...ϕ a ϕ b |=|ab|
(1.31)
There are two possible spin eigenfunctions, singlet and triplet, with S equal to 0 and
1, respectively. The projection operators are directly obtained from Eq. 1.30
ˆ
P 0 = ˆ
S
2 − 2
ˆ
P 1 = ˆ
S
2 − 0
(1.32)
The result of applying ˆ
S 2 on Φ 1 is given in Eq. 1.27, and hence, the projection
operators give
ˆ
P 0 |ab|=( ˆ
S
2 − 2)|ab|=|ab|+|ab|−2|ab|=|ab|−|ab|
(1.33a)
ˆ
P 1 |ab|=( ˆ
S
2 − 0)|ab|=|ab|+|ab|
(1.33b)
The functions have to be multiplied by
1
√
2
to obtain the properly normalized expressions.
1.6 (a) Find the other two components of the triplet spin eigenfunctions by
applying the ladder operators on the M S = 0 component of the triplet function.
(b) Derive the singlet and triplet spin eigenfunctions by projection using Φ 2
of Eq. 1.31.
1.2.2 Spin Functions by Diagonalization
One way to find the eigenvalues and eigenvectors of an operator is to diagonalize
the matrix representation of the operator in a complete basis. Therefore, a natural
alternative to the projection method is the process of diagonalizing the matrix representation of the ˆ
S 2 operator. The basis of the matrix representation is formed by
the individual determinants. The resulting eigenvectors are the spin eigenfunctions
(linear combinations of these basis functions, the determinants) and the corresponding eigenvalues indicate the spin of the eigenfunction. The method is straightforward
in its application but can require a substantial amount of analytical work since all
matrix elements of ˆ
S 2 are needed, which can become rather cumbersome for systems
with an elevated number of unpaired electrons. The method is illustrated for a system