**Spin Contamination** The determinant |a

__b__| is not an eigenfunction of S^2 but the following linear combinations are:

|a

__b__|+|b

__a__|= (ab+ba)(αß-ßα) (Ms=0 ; Singlet)

|a

__b__|-|b

__a__|= (ab-ba)(αß+ßα) (Ms=0 ; Triplet)

Calculating the expectation value of S^2 for these 2 determinantal wavefunctions:

<|a

__b__|+|b

__a__| S^2 |a

__b__|+|b

__a__|>=

<(ab+ba)(αß-ßα) |S^2| (ab+ba)(αß-ßα)> =

<(ab+ba) | (ab+ba)><(αß-ßα) |S^2| (αß-ßα)> =

<(αß-ßα) |S^2| (αß-ßα)> =

<(αß-ßα) |0 (αß-ßα)> = 0 = S(S+1) => S=0

<|a

__b__|-|b

__a__| S^2 |= |a

__b__|-|b

__a__|> =

<(ab-ba)(αß+ßα) |S^2| (ab-ba)(αß+ßα)> =

<(ab-ba) | (ab-ba)> <(αß+ßα) |S^2| (αß+ßα)> =

<(αß+ßα) |S^2| (αß+ßα)> =

<(αß+ßα) |2 (αß+ßα)> = 2 = S(S+1) => S=1

In short:

<(αß-ßα) |S^2| (αß-ßα)> = 0

<(αß+ßα) |S^2| (αß+ßα)> = 2

Now we can write the single determinant |a

__b__| as being half singlet and half triplet:

|a

__b__| = ½|S> + ½|T> =

½ (|a

__b__|+|b

__a__|) + ½(|a

__b__|-|b

__a__|) =

½|a

__b__|+½|b

__a__| + ½|a

__b__|-½|b

__a__| = |a

__b__|

**Edited: **The factor should be 1/sqrt(2)

The orbitals a and b can be seen as average orbitals for the singlet and the triplet.

When we write the single determinant as half singlet and half triplet, S^2 could also be half the singlet value and half triplet value. Then this is not a proper expectation value (would it be an average value?) and the value is 1.

The expectation value of S^2 for a broken symmetry solution <BS|S^2|BS> where alpha and beta spins are described by other orbital spaces, can have different values as listed above. Orbitals from the alpha orbital space and those from the beta orbital space are not necessary orthogonal. This is the source for spin contamination.