Development note  

Changes to this chapter were last made on 11 April 2005. 
If
Even in the case of atoms, finding a set of commuting operators inevitably involves some level of approximation. Case and point being the Dirac equation describing a particle in a central field―to which the Schrödinger equation is an approximation. The relativistic Hamiltonian operator can be written as
Both
Similarly, for the spin operator we have
Hence a set of commuting observables do exist for a relativistic particle; they are
We will attempt to follow international convention [see Appendix A], in which
case the operators
respectively.
For a manyelectron atomic Hamiltonian that neglects electronelectron interactions, that is,
for a fictitious system of independent electrons described by
Eqs. (3.1)(3.4), we can write the atomic orbitals as
The solutions are the hydrogenlike wavefunctions,
However, if the
and the hydrogenlike wavefunctions will be useful in constructing only approximate manyelectron wavefunctions. Nevertheless, because of the spherical symmetry of atoms, the total orbital angular momentum, in the nonrelativistic limit, is a constant of the motion [also see Bernath, pp. 131132],
where the zcomponent of singleparticle orbital angular momentum operator is given by
and we have used the following derivatives
Replacing
So, in fact, we see from Eq. (4.29) that when electronelectron interactions are included in the Hamiltonian
of a manyelectron atom
As mentioned previously, the total orbital angular momentum,
Eq. (4.31) is equal to the commutator of the total orbital angular momentum operator and the Hamiltonian operator
The singleelectron orbital angular momentum eigenvalue equations were given in Eqs. (4.19) and (4.20). Similarly, the total orbital angular momentum eigenvalue equations of multielectron atoms are
For Nelectrons we can iterate this process by first combining the l values for two electrons, as was done above [Liboff, pp. 354355]
then combining the
and similarly for the fourth electron
Finally, for Nelectrons we have
Table 4.1. Orbital angular momentum quantum numbers and symbols used to describe atoms and molecules.
In Equations (4.4)(4.8), we showed that
An atomic energy level can be labeled with the term symbol
which will be discussed in detail later.
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Constructing a symmetric or antisymmetric state using a linear combination of base kets from the decoupled
representation [Eq. (2.88)] will not necessarily produce an eigenstate of the angular momentum
operators. This means that a single Slater determinant will not be sufficient to construct an eigenstate with the same symmetry properties of
the exact wavefunction. In Eqs. (4.25)(4.29) we showed that if electronelectron interactions are included in the Hamiltonian, then
In Eq. (4.48) we have assumed that j odd labels a spinorbital occupied by an electron with its spinup, and j even labels an orbital occupied by an electron with its spindown. This convention allows us to write
where the plus sign is for j odd, the minus sign is for j even. The choice of i = 1 as an example in Eqs. (4.48) and (4.49) was arbitrary. It has also been assumed that the spinorbitals are as defined in Eq. (3.28).
From Eq. (4.48) it can be seen that for the manyelectron wavefunction
(
where
is the total number of electrons. One could have also derived Eq. (4.50) by performing a sum over all the electrons in Eq. (4.48), in which case it would have found that
and when
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Spinorbitals constructed from hydrogenlike wavefunctions satisfy
Eqs. (4.56) and (4.57).
In general, however, the manyelectron wavefunction
(
The fact that Slater determinants will not be eigenstates of singleelectron operators, even without electronelectron interactions, can be seen by applying
If the eigenvalue of the operator
If each spinorbital used in the construction of a Slater determinant is an eigenstate of the oneelectron angular momentum operators, as in Eqs. (4.56) and (4.57), then a single Slater determinant will be an eigenstate of
and when
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and in atomic units (a.u.) it is
where the oneelectron Hamiltonians are
and the electronelectron interaction energy is
If the electronelectron interactions are neglected, then the exact solution can be constructed from the hydrogenlike orbitals,
With the four spinorbitals, we can construct six Slater determinants, or configuration functions, corresponding to the diagrammatic representations shown in Figure 4.1
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As we showed in Section 4.2, all six of the Slater determinants are eigenfunctions of
Ψ_{0} and Ψ_{5}  S = 0  M_{S} = 0 
Ψ_{1}  S = 1  M_{S} = 1 
Ψ_{4}  S = 1  M_{S} = 1 
The configuration functions, Ψ_{2} and Ψ_{3}, however, are not eigenfunctions of
where we have used the fact that
where, again, we have used the fact that
we see that the sum, Ψ_{2} + Ψ_{3}, is an eigenstate in which S(S + 1 ) = 1 (1 + 1), and therefore S = 1. By subtracting Eq. (4.77) from (4.76)
and the spin singlet is equal to
Thus, all six of the spin eigenstates constructed from the twoelectron Slater determinants can be written as a product of a spatial function and a spin function. This manifestly simple decoupling of the spin degrees of freedom in the twoelectron atom is not true of manyelectron systems in general. The spinangular momentum quantum numbers of the six twoelectron Slater determinants are summarized in Table 4.2
Table 4.2. The spinangular momentum quantum numbers of twoelectron wavefunctions constructed from six twoelectron Slater determinants.
Ψ  S  M_{S}  

Ψ_{0}, Ψ_{5},
 0  0  
Ψ_{1}  1  1  
 1  0  
Ψ_{4}  1  1 
To this point, the two sets of quantum numbers n_{1}, l_{1}, m_{l1}, and n_{2}, l_{2}, m_{l2} have not been specified. In Table 4.3, the electron configurations of six twoelectron Slater determinants are shown for different values of the quantum numbers n_{1}, l_{1}, n_{2} and l_{2}. Note that the Slater determinants Ψ_{1}, Ψ_{2}, Ψ_{3}, and Ψ_{4} all have the same electron configurations. The electron configuration does not specify the spin quantum numbers, or the zcomponent of the orbital angular momentum. Thus, there may be several different states corresponding to a given electron configuration.
Table 4.3. The electron configurations of six twoelectron Slater determinants for different values of the quantum numbers n_{1}, l_{1}, n_{2} and l_{2}. Note that the Slater determinants Ψ_{1}, Ψ_{2}, Ψ_{3}, and Ψ_{4} all have the same electron configurations.
n_{1} l_{1}  n_{2} l_{2}  Ψ_{0}  Ψ_{1}  Ψ_{2}  Ψ_{3}  Ψ_{4}  Ψ_{5} 

1 0  1 0  1s^{2}  1s^{2}  1s^{2}  1s^{2}  1s^{2}  1s^{2} 
1 0  2 0  1s^{2}  1s 2s  1s 2s  1s 2s  1s 2s  2s^{2} 
1 0  2 1  1s^{2}  1s 2p  1s 2p  1s 2p  1s 2p  2p^{2} 
1 0  3 0  1s^{2}  1s 3s  1s 3s  1s 3s  1s 3s  3s^{2} 
1 0  3 1  1s^{2}  1s 3p  1s 3p  1s 3p  1s 3p  3p^{2} 
1 0  3 2  1s^{2}  1s 3d  1s 3d  1s 3d  1s 3d  3d^{2} 
1 0  4 0  1s^{2}  1s 4s  1s 4s  1s 4s  1s 4s  4s^{2} 
1 0  4 1  1s^{2}  1s 4p  1s 4p  1s 4p  1s 4p  4p^{2} 
1 0  4 2  1s^{2}  1s 4d  1s 4d  1s 4d  1s 4d  4d^{2} 
1 0  4 3  1s^{2}  1s 4f  1s 4f  1s 4f  1s 4f  4f^{2} 
2 0  2 0  2s^{2}  2s^{2}  2s^{2}  2s^{2}  2s^{2}  2s^{2} 
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We can use the orbital angular momentum ladder operators
to establish the orbital angular momentum properties of the twoelectron Slater determinants in the same as was done for spin in Eqs. (4.76)  (4.79). It is useful to realize that the determinants Ψ_{1}, Ψ_{2}, Ψ_{3}, and Ψ_{4} have the same orbital properties, and Ψ_{0} and Ψ_{5} can be obtained from Ψ_{2} or Ψ_{3} by setting
To obtain
If we define four new Slater determinants
then Eq. (4.84) can be written as
Using Eq. (4.82), together with Eq. (4.87) and
Next, we will find cases in which the Slater determinant Ψ_{1}, defined by Eq. (4.71) and
Figure 4.1(b), is an eigenstate of
Table 4.4. Four cases in which the twoelectron Slater determinant discussed in the text is an orbital angular momentum eigenstate.
l_{1}  l_{2}
 L  M_{L} 

l_{1} l_{1}  l_{2} l_{2}  l_{1} + l_{1}  l_{1} + l_{1} 
l_{1}  l_{1}  l_{2}  l_{2}  l_{1} + l_{1}   l_{1}  l_{1} 
0 0  l_{2}
 l_{2}  
l_{1}
 0 0  l_{1} 
It was all ready mentioned in Section 4.4.2 that the example of a twoelectron atom is unique in that the antisymmetric eigenstates of
is the spin part of the wavefunction. The minus sign − in the S = 0 function, Ψ_{00}, is highlighted to emphasize that it is the only antisymmetric spin function of the four in Eq. (4.91). The spin triplet functions, Ψ_{11}, Ψ_{10}, and Ψ_{11}, are all symmetric.
The spatial part of the wavefunction, for the first three cases in Table 4.4, is
where the plus sign in Eq. (4.92), +, corresponds to a symmetric
From the triangle rule, we know that if l_{1} = 0, then the only possible value of L is L = l_{2}, and all the eigenstates of
In the last term of Eq. (4.93), we have adopted the notation of Slater, in which the determinants are represented by their microstates [Atkins, p. 240; Lowe, p. 153; Bernath, p. 133], or the product of the diagonal elements of the Slater determinant, without the
To give a more specific example, let l_{1} = 0 and l_{2} = 1, corresponding to a sp configuration. When l_{1} = 0 and l_{2} = 1, the determinantal wavefunction Ψ_{1} becomes
together with the oneelectron ladder operators in Eqs. (2.15) and (2.16), one can obtain
and as a reminder, Ψ_{1}, Ψ_{2}, and Ψ_{3} are defined by Eqs. (4.71)  (4.73) and Figure 4.1. From Eq. (4.96), it is easy to see that
and the application of the ladder operator even provides the correct normalization factor.
Similarly, for the orbital angular momentum we can use
together with the oneelectron ladder operators in Eq. (2.91), with the substitutions
All the spin and orbital angular momentum eigenstates of the configuration sp, shown below in Table 4.5, can be obtained by following this kind of approach. For more complicated examples, other methods, such as the use of projection operators, may be more appropriate [SlaterII, p. 79].
Table 4.5. Spin and orbital angular momentum eigenstates of the configuration sp.
^{2S + 1}P  M_{L}  M_{S}  

^{3}P  1  1  
1  0  
1  1  
0  1  
0  0  
0  1  
1  1  
1  0  
1  1  
^{1}P  1  0  
0  0  
1  0 
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In the last section [Section 4.4.3], we listed in Table 4.5 all the spin and orbital angular momentum eigenstates of the configuration sp in the decoupled representation. The ClebschGordan coefficients for an L = 1 orbital angular momentum vector, and a S = 1 spin angular momentum vector, shown in Table 4.6, can be used used to construct eigenstates in the coupled representation. Any set of states that are eigenstates of
It can be seen from Table 4.5, that the sp configuration consists of one L = 1, S = 1, eigenspace spanned by
(2L + 1)(2S + 1) = 9 basis vectors, and a L = 1, S = 0, eigenspace spanned by (2L + 1)(2S + 1) = 3 basis vectors. Representations in these two vector spaces can be expressed symbolically, in group theory notation, as
Table 4.6. The ClebschGordan coefficients for L = 1, S = 1. The three possible values of J are J = 0, 1, 2, and the corresponding term symbols are ^{3}P_{0}, ^{3}P_{1}, and ^{3}P_{2}, respectively.
J = 2  M_{J} = 2  1  0  0  0  0  0  0  0  0  
M_{J} = 1  0  0  0  0  0  0  0  
M_{J} = 0  0  0  0  0  0  0  
M_{J} = 1  0  0  0  0  0  0  0  
M_{J} = 2  0  0  0  0  0  0  0  0  1  
J = 1  M_{J} = 1  0  0  0  0  0  0  0  
M_{J} = 0  0  0  0  0  0  0  0  
M_{J} = 1  0  0  0  0  0  0  0  
J = 0  M_{J} = 0  0  0  0  0  0  0 
Table 4.6 can also be written in matrix form as
On the other hand, if S = 0, then
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The change of basis between the RussellSaunders coupling scheme and the jjcoupling scheme can be obtained by calculating the matrix elements of
Table 4.7. The transformation coefficients for the change of basis between the RussellSaunders coupling scheme and the jjcoupling scheme for a sl configuration.



Table 4.8. By setting l = 1, one obtains the transformation coefficients for the change of basis between the RussellSaunders coupling scheme and the jjcoupling scheme for a sp configuration.



If we set l = 1 in Table 4.7, then we obtain the set of coefficients in Table 4.8, and these are the ones that will be used in the following example. Furthermore, the eigenkets in the jjcoupling scheme can be written as
It is easy to see from Eq (4.106) how, in general, states with different values of L and different values of S are coupled together to form eigenkets in the jjcoupling scheme.
...to be continued. 
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