Note This appendix contains details of some of the equations presented in the text. Changes were last made on 25 March 2005.

## B.1.  $〈\Psi |{\stackrel{^}{S}}^{2}|\Psi 〉={\hslash }^{2}S\left(S+1\right)$   and other details.

Since a single Slater determinant is always an eigenstate of   ${\stackrel{^}{S}}_{z}$,   we can focus our attention on understanding what effect the operators   ${\stackrel{^}{S}}_{\mp }{\stackrel{^}{S}}_{±}$   have on a Slater determinant. It will be sufficient to show that when the ladder operators  (${\stackrel{^}{S}}_{±}$)  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   ${\stackrel{^}{S}}_{±}|\Psi 〉=0$,   since, in general, for an operator   $\stackrel{^}{A}$

and   $〈\Psi |\Psi \mathbf{\text{'}}〉=\lambda$   does not necessarily imply that   $|\Psi 〉$   is an eigenstate of   $\stackrel{^}{A}$   (i.e., $|\Psi \text{'}〉=\lambda |\Psi 〉$),  where   $\lambda$   is a constant. In fact,   $\Psi$   will not be an eigenstate of   ${\stackrel{^}{S}}_{\mp }{\stackrel{^}{S}}_{±}$   and   ${\stackrel{^}{\mathbf{S}}}^{2}$   if   $\Psi$   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   ${\stackrel{^}{S}}_{\mp }{\stackrel{^}{S}}_{±}$   will be written as the sum of a one-particle operator   $\stackrel{^}{U}$   and a two-particle operator   $\stackrel{^}{V}$, 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   $〈\Psi |\stackrel{^}{U}|\Psi 〉$,   and Eq. (5.14), times a factor of two (since   ${\stackrel{^}{s}}_{i\mp }{\stackrel{^}{s}}_{j±}\text{\hspace{0.17em}and\hspace{0.17em}}{\stackrel{^}{s}}_{j\mp }{\stackrel{^}{s}}_{i±}$   must both be included in the summation) to evaluate   $〈\Psi |\stackrel{^}{V}|\Psi 〉$;   this gives us

In our previous discussions of spin, we used the notation of Eq. (3.30), which maps a spatial wavefunction, ψi, to two spin-orbitals, χ2i-1 and χ2i. 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   $|{\psi }_{i}〉=|{n}_{i}{l}_{i}{m}_{{l}_{i}}〉$, and we have introduced the superscript label (ms) so the discussion that follows will be valid for orbital angular momentum with the appropriate definition of the spin-orbital indices and the substitutions   ${\chi }_{i}^{\left({m}_{s}\right)}\to {\chi }_{i}^{\left({m}_{l}\right)}$,     $\stackrel{^}{\mathbf{S}}\to \stackrel{^}{\mathbf{L}}$, etc. The superscript labels, (ms) and (ml), only serve to remind us whether the spin-orbital indices are defined to have a one-to-one correspondence with the quantum number ms (or in the next section ml). 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   $\left({m}_{s}=-\frac{1}{2}\right)$,   j even labels a spin-up state   $\left({m}_{s}=+\frac{1}{2}\right)$,   and the ladder operators have the same effect on the spin-orbital subscript as the quantum number ms (or in the next section, ml). 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

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   $〈\Psi \text{}|{\stackrel{^}{S}}^{2}|\Psi 〉=0$,   as expected.

If   $\Psi$   includes open shells, then Eq. (B.16) is equal to zero if   ${\chi }_{i}^{\left({m}_{s}\right)}\text{\hspace{0.17em}or\hspace{0.17em}}{\chi }_{j}^{\left({m}_{s}\right)}$   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   ${\stackrel{^}{\mathbf{S}}}^{2}$   can be calculated using

where

and

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

where

and No 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   $\Psi$   will not be an eigenstate of   ${\stackrel{^}{S}}_{\mp }{\stackrel{^}{S}}_{±}$   and, therefore, neither will it be an eigenstate of   ${\stackrel{^}{\mathbf{S}}}^{2}$.