To better understand the effect   S ^ 2   has on Slater determinants, it will be useful to consider   S ^ 2   written in terms of the ladder operators [Eq. (2.5)]

Since a single Slater determinant is always an eigenstate of   S ^ z ,   we can focus our attention on understanding what effect the operators   S ^ ∓ S ^ ±   have on a Slater determinant. It will be sufficient to show that when the ladder operators  ( S ^ ± )  are applied to a closed shell Slater determinant, the result is zero. But first, it will be shown that

which is a weaker statement than   S ^ ± ⁢ | Ψ 〉 = 0 ,   since, in general, for an operator   A ^  

and   〈 Ψ ⁢ | Ψ ' 〉 = λ   does not necessarily imply that   | Ψ 〉   is an eigenstate of   A ^   (i.e.,  | Ψ ' 〉 = λ | Ψ 〉 ),  where   λ   is a constant. In fact,   Ψ   will not be an eigenstate of   S ^ ∓ ⁢ S ^ ±   and   S ^ 2   if   Ψ   is an open shell Slater determinant in which the spins of the open shell electrons are not aligned.

To prove Eq. (B.2), the operator   S ^ ∓ ⁢ S ^ ±   will be written as the sum of a one-particle operator   U ^   and a two-particle operator   V ^ , i.e.,

where

The expectation value of Eq. (B.4) with respect to a many-electron state is given by

Next, we can use Eq. (5.10) to evaluate   〈 Ψ | ⁢ U ^ ⁢ | Ψ 〉 ,   and Eq. (5.14), times a factor of two (since   s ^ i ∓ ⁢ s ^ j ±  and  s ^ j ∓ ⁢ s ^ i ±   must both be included in the summation) to evaluate   〈 Ψ | ⁢ V ^ ⁢ | Ψ 〉 ;   this gives us

In our previous discussions of spin, we used the notation of Eq. (3.30), which maps a spatial wavefunction, ψ_i, to two spin-orbitals, χ_2i-1 and χ_2i. With this definition of the spin-orbital labels there is a one-to-one correspondence between spin-orbital label and the z-projection of the angular momentum quantum number, and the spin-up orbital follows the spin-down orbital sequentially for each spatial wavefunction. In this appendix, we will redefine the spin-orbital labels so our discussion can be easily generalized to orbital angular momentum. In ket notation, our restricted spin-orbitals are defined by

where   | ψ i 〉 = | n i l i m l i 〉 , and we have introduced the superscript label (m_s) so the discussion that follows will be valid for orbital angular momentum with the appropriate definition of the spin-orbital indices and the substitutions   χ i ( m s ) → χ i ( m l ) ,     S ^ → L ^ , etc. The superscript labels, (m_s) and (m_l), only serve to remind us whether the spin-orbital indices are defined to have a one-to-one correspondence with the quantum number m_s (or in the next section m_l). In the case of spin angular momentum, the redefinition is trivial

The main difference between this notation and the notation introduced in Chapter 2 [see Eqs. (2.29), (3.30), and (3.31)] is that here j odd labels a spin-down state   ( m s = - 1 2 ) ,   j even labels a spin-up state   ( m s = + 1 2 ) ,   and the ladder operators have the same effect on the spin-orbital subscript as the quantum number m_s (or in the next section, m_l). The effect the single-particle ladder operators have on the spin eigenstates can be summarized by the following equation

Since we are dealing with spin-half particles, Equation (B.12) can be written as

Equation (B.13) can be used to evaluate the matrix elements in Eqs. (B.7) and (B.8). The matrix elements of the single-particle operator in Eq. (B.7) are

where we have used the fact that   S ^ - = S ^ + † . The matrix elements of the two-particle operator in Eq. (B.8) are

and

If we substitute Eqs. (B.14), (B.15), and (B.16) into Eqs. (B.7) and (B.8), we obtain the following for a restricted closed-shell Slater determinant

Finally, from Eqs. (B.6) and (B.17) it is easy to see that for a restricted closed-shell Slater determinant we have

and hence   〈 Ψ | S ^ 2 | Ψ 〉 = 0 ,   as expected.

If   Ψ   includes open shells, then Eq. (B.16) is equal to zero if   χ i ( m s )  or  χ j ( m s )   corresponds to an open shell spin-orbital, and therefore the contribution to Eq. (B.8) from open shells is zero. Furthermore, the sum in Eqs. (B.7) can be split into two sums: one over closed shell orbitals and one over open-shell orbitals. But it was just shown that the net contribution from closed shells is zero, therefore only the open shells contribute to the total spin angular momentum of a many-electron wavefunction.

It is simple to repeat the above the above discussion to include open shell Slater determinants. The matrix elements of the operator   S ^ 2   can be calculated using

where

and

By combining Eqs. (B.20), (B.21), and (B.22) one obtains

If all the open shell electrons have the same spin, then   m s i = ± s ,   and Eq. (B.23) becomes

where

and N_o is the number of open shell electrons,   N ↑   is the total number of spin-up electrons, and   N ↓   is the total number of spin-down electrons. If the spins of the open shell electrons are not aligned, then   Ψ   will not be an eigenstate of   S ^ ∓ ⁢ S ^ ±   and, therefore, neither will it be an eigenstate of   S ^ 2 .