D.2 Extended Example: The He Atom
633
electronic spin states form spectroscopically distinct species. This distinctness was
acknowledged in the early part of the twentieth century by referring to the set of
metastable triplet terms as ortho-helium (meaning high-spin) and the set of singlet
terms as para-helium.
At the third (and final stage for this discussion), the spin–orbit interaction is taken
into account by adding the term H so = A S · L to the Hamiltonian. It can be shown
that S· L commutes neither with L, nor with S, but it does commute with their vector
sum J = L + S. The good quantum numbers are now only J, M J , corresponding
to the operators
J 2 ,
J z (however, we retain L, S so that we know which level is the
origin for a particular value of J appearing in our labelling procedure). The quantum
number J has the set of values determined by J = L+S, L+S −1, · · · , |L−S|.
The J values characterize what are referred to as levels, and each pair of (J, M J )values designates a specific state within a level. The value of J is typically given
as a subscript to the term symbol from which it was deduced. Thus, for the two
examples given at the previous stage of our description, namely the 1 D and 3 P
terms, we would have levels designated by 1 D 2 in the first case (because there is
only one possible value for J when L = 2 and S = 0), and 3 P 2 , 3 P 1 , 3 P 0 arising
from the original 3 P term. Hence, we see that the ‘singlet D’ term indeed has only
one (fivefold degenerate) level, while the ‘triplet P ’ term gives rise to three levels,
with degeneracies 5, 3, and 1, respectively. For atoms, the term symbol tells us the
degeneracy of the energy level to which the symbol corresponds: this degeneracy is
always given by the value of 2J + 1.
For atomic configurations with nonequivalent valence electrons, such as the
excited electronic (1s) 2 (2s) 1 (2p) 1 configuration for Be, which has a total of 12
possible states (consisting of all possible combinations of the two available states
for the 2s orbital and the six available states for the 2p orbital), all possible terms
are present (as the Pauli Principle has already been satisfied once the electrons
have different values for their azimuthal quantum numbers). The example of the
Be excited electronic configuration Be[(1s) 2 (2s) 1 (2p) 1 ] serves to illustrate this
point. We have l 1 = 0 and l 2 = 1, which gives L = l 1 + l 2 = 1 as the only
total orbital angular momentum possible, and this, in turn, leads us to P terms.
Each electron has spin
1
2 , so that the total electron spin S can have values 1 and
0, which leads to 3 P and 1 P terms. For the triplet term, we have L = 1, S = 1,
from which we obtain J = 2, 1, 0, while for 1 P , we have L = 1 and S = 0,
so that J = 1 only. Thus, we arrive finally at a 1 P 1 level, with 3 states, plus the
3 P 2 , 3 P 1 , 3 P 0 levels arising from the 3 P term, with 9 states (made up of 5, 3, 1
state(s), respectively, for the three levels) for a total of 3 + 9 = 12 total electronic
states for this configuration. This example also illustrates the principle that the
number of states available to an atom in a given configuration remains the same
even when the various electronic coupling mechanisms are taken into account. This
principle can be utilized as a check on the final results obtained by the Russell–
Saunders coupling scheme employed in the present discussion or, indeed, for any
other coupling scheme that may be necessitated.
633
electronic spin states form spectroscopically distinct species. This distinctness was
acknowledged in the early part of the twentieth century by referring to the set of
metastable triplet terms as ortho-helium (meaning high-spin) and the set of singlet
terms as para-helium.
At the third (and final stage for this discussion), the spin–orbit interaction is taken
into account by adding the term H so = A S · L to the Hamiltonian. It can be shown
that S· L commutes neither with L, nor with S, but it does commute with their vector
sum J = L + S. The good quantum numbers are now only J, M J , corresponding
to the operators
J 2 ,
J z (however, we retain L, S so that we know which level is the
origin for a particular value of J appearing in our labelling procedure). The quantum
number J has the set of values determined by J = L+S, L+S −1, · · · , |L−S|.
The J values characterize what are referred to as levels, and each pair of (J, M J )values designates a specific state within a level. The value of J is typically given
as a subscript to the term symbol from which it was deduced. Thus, for the two
examples given at the previous stage of our description, namely the 1 D and 3 P
terms, we would have levels designated by 1 D 2 in the first case (because there is
only one possible value for J when L = 2 and S = 0), and 3 P 2 , 3 P 1 , 3 P 0 arising
from the original 3 P term. Hence, we see that the ‘singlet D’ term indeed has only
one (fivefold degenerate) level, while the ‘triplet P ’ term gives rise to three levels,
with degeneracies 5, 3, and 1, respectively. For atoms, the term symbol tells us the
degeneracy of the energy level to which the symbol corresponds: this degeneracy is
always given by the value of 2J + 1.
For atomic configurations with nonequivalent valence electrons, such as the
excited electronic (1s) 2 (2s) 1 (2p) 1 configuration for Be, which has a total of 12
possible states (consisting of all possible combinations of the two available states
for the 2s orbital and the six available states for the 2p orbital), all possible terms
are present (as the Pauli Principle has already been satisfied once the electrons
have different values for their azimuthal quantum numbers). The example of the
Be excited electronic configuration Be[(1s) 2 (2s) 1 (2p) 1 ] serves to illustrate this
point. We have l 1 = 0 and l 2 = 1, which gives L = l 1 + l 2 = 1 as the only
total orbital angular momentum possible, and this, in turn, leads us to P terms.
Each electron has spin
1
2 , so that the total electron spin S can have values 1 and
0, which leads to 3 P and 1 P terms. For the triplet term, we have L = 1, S = 1,
from which we obtain J = 2, 1, 0, while for 1 P , we have L = 1 and S = 0,
so that J = 1 only. Thus, we arrive finally at a 1 P 1 level, with 3 states, plus the
3 P 2 , 3 P 1 , 3 P 0 levels arising from the 3 P term, with 9 states (made up of 5, 3, 1
state(s), respectively, for the three levels) for a total of 3 + 9 = 12 total electronic
states for this configuration. This example also illustrates the principle that the
number of states available to an atom in a given configuration remains the same
even when the various electronic coupling mechanisms are taken into account. This
principle can be utilized as a check on the final results obtained by the Russell–
Saunders coupling scheme employed in the present discussion or, indeed, for any
other coupling scheme that may be necessitated.