Appendix B. More information about the equations presented in Chapter 3.


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   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.,


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, 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   | ψ 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   χ i ( m s ) χ i ( m l ) ,     S ^ 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   ( 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 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

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


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



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


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   Ψ   will not be an eigenstate of   S ^ S ^ ±   and, therefore, neither will it be an eigenstate of   S ^ 2 .  

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   χ j ( m s ) ( x ) ,   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

As an example, we will first show that   S ^ + Ψ = 0   for the simple case of a two-electron restricted closed-shell Slater determinant, and then generalize the discussion to include any single restricted Slater determinant. The two-electron closed-shell Slater determinant is given by

where in the last term we have used the short-hand notation   i = x i   to label the coordinate of the ith electron. If we apply the single-electron ladder operators,   s ^ 1 +   and   s ^ 2 + ,   to the two-electron Slater determinant, we obtain


In this example,   S ^ +   is just the sum of the two single-electron ladder operators, and thus

To show that   S ^ ± Ψ = 0   in general, first consider Eq. (3.17) with   s ^ i ±   substituted for the arbitrary operator   A ^ i  

Equation (B.32) will be used to show how terms cancel each other when   S ^ ±   is applied to a restricted closed-shell determinant. But first we will apply   s ^ 1 ±   to an N-electron Slater determinant

Now it is easy to see from Eq. (B.33) and Eq. (B.32) that if   l = k ± 1 ,  then

Similarly, if   m = k ± 1 ,   then

More generally, if   n = k ± 1 ,   then

In other words, for every value of k, either   [ 1 ( - 1 ) k ] = 0 ,   or there is a value of   j 1   such that

The sum over all electrons in   S ^ ± = i = 1 N s ^ i ±   ensures that the cancellation occurs for every value of k, and the choice of   s ^ 1 ±   was arbitrary, therefore   S ^ ± Ψ = 0   for a restricted closed-shell determinant.

It is easy to extend the proof that S ^ ± Ψ = 0   for a closed-shell Slater determinant to the orbital angular momentum. In fact, the results of Eqs. (B.32)-(B.37) only require a couple minor changes to show that L ^ ± Ψ = 0 for the orbital angular momentum. First, note that, besides the superscript label (ms) 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 ml. So instead of Eq. (B.12), we would have

and instead of Eq. (B.33) we have

The subscript k on   χ k ( m l )   labels a unique spin-orbital, and therefore it represents a set of quantum numbers for that spin-orbital. The only requirement for the proof to be valid is that for every spin-orbital   χ k ( m l ) ,   there is another spin-orbital   χ j ( m l )   included in the Slater determinant, such that   j = k ± 1 , or   ( l k m lk ) ( l k ± m lk + 1 ) = 0 .   It isn't possible to express a mapping of the orbital angular momentum quantum numbers to the spin-orbital indices as concisely as Eq. (3.30) [or Eq. (B.10) and (B.11) ], but we can certainly redefine the labeling of the spin-orbitals   χ k ( m l )   such that Eq. (B.38) is true. For 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

l ^ + | χ 1 ( m l ) = 2 | χ 2 ( m l )
l ^ + | χ 2 ( m l ) = 2 | χ 3 ( m l )
l ^ + | χ 3 ( m l ) = 0

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   χ i ( m s ) χ i ( m l ) ,     s ^ i ± l ^ i ± ,   and

The arguments used to show that   S ^ ± Ψ = 0   and   S ^ 2 Ψ = 0   for closed shell determinants are also valid for orbital angular momentum. Therefore   L ^ ± Ψ = 0   and   L ^ 2 Ψ = 0   for single closed shell Slater determinants.

We can also repeat the discussion that lead to Eq. (B.25) for orbital angular momentum by simply making the substitutions   S ^ L ^ ,     s ^ i ± l ^ i ± ,     m s m l , and   s l , etc. Equation (B.23) then becomes

Similar to the case of spin angular momentum, if   m l i = l i ,   for all the open shell electrons, or   m l i =- l i for all the open shell electrons,   then Eq. (B.55) becomes


Single Slater determinants with extremum values of S, L, MS and ML, given by Eqs. (B.25) and (B.57), will be simultaneous eigenstates of   L ^ 2 ,  S ^ 2 ,  J ^ 2 ,  L ^ z ,  S ^ z , and  J ^ z .  The states with the maximum values of S and L, and intermediate values of MS and ML, i.e.,   M L = - L , - ( L + 1 ) , , ( L - 1 ) , L ,  and   M S = - S , - ( S + 1 ) , , ( S - 1 ) , S , are easily obtained by using the ladder operators   L ^ ±   and   S ^ ± , as will be demonstrated in Chapter 4.