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

To better understand the effect

which is a weaker statement than
*in general*, for an operator

To prove Eq. (B.2), the operator

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

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

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*
*j even labels a spin-up state*
*m _{s}* (or in the next section,

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

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

If

It is simple to repeat the above the above discussion to include open shell Slater determinants. The matrix elements of the operator

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

If all the open shell electrons have the same spin, then

As a reminder, the Slater determinant representation of Eq. (3.36) can be written as

If the discussion is limited to restricted spin-orbitals, and, as before in Eq. (3.30), for a spin-orbital
*j odd* labels a spin eigenstate occupied by an electron with its *spin-down*, and *j even* labels a spin-orbital occupied by an electron with its *spin-up*. The effect the single-particle ladder operators have on the spin-orbitals can be summarized by the following equation

In this example,

To show that

Equation (B.32) will be used to show how terms cancel each other when
*N*-electron Slater determinant

Now it is easy to see from Eq. (B.33) and Eq. (B.32) that if

Similarly, if

It is easy to extend the proof that
*m _{s}*) we assigned to the spin-orbitals, the quantum numbers in Eq. (B.33) are not shown explicitly. We could have just as well defined the spin-orbital labels relative to the orbital angular momentum quantum number

and instead of Eq. (B.33) we have

The subscript *k* on
*l* = 0 spin-orbitals, the mapping can be written as

and Eq. (B.38) is true trivially

For *l* = 1 spin-orbitals, the mapping is

and for *l* = 1 subshells we can also write the equivalent of Eq. (B.27) as

It is easy to verify that Eq. (B.45) gives the correct results. For example, with *k* = 1, 2, 3, we have

The mapping for *l* = 2 spin-orbitals is

and so on for spin-orbitals with *l* = 3, 4, 5,..., etc.

With this definition of spin-orbital indices, we can see that Eqs. (B.32) through (B.37) will be true for orbital angular momentum with the substitutions

We can also repeat the discussion that lead to Eq. (B.25) for orbital angular momentum by simply making the substitutions

Similar to the case of spin angular momentum, if

Single Slater determinants with extremum values of *S*, *L*, *M _{S}* and