2.2 The Eigenstates of Many-Electron Atoms
35
field, for the time being neglecting any spin-orbit coupling. The total orbital angular
moment operators ˆ
L 2 , ˆ
L z and the total spin angular moment operators ˆ
S 2 and ˆ
S z commute with the Hamilton operator ˆ
H where each electron moves in a field of spherical
symmetry due to the nucleus and the field due the other electrons. Therefore, the
eigenfunctions of ˆ
H are in general also eigenfunctions of the other four operators.
Only in the case of degenerate eigenstates of ˆ
H one may choose (or find) eigenfunctions that are not simultaneously eigenfunctions of the other four operators. In that
case, however, the eigenfunctions of ˆ
H may always be rotated within the degenerate
set to become also eigenfunctions of the other operators.
This implies that these eigenfunctions of ˆ
H can be labelled using the quantum
numbers S, M S , L and M L . The energy eigenvalues only depend on the eigenvalues
of ˆ
L 2 and ˆ
S 2 , and not on M S and M L . Therefore, the degenerate set of eigenfunctions
of the free-atom Hamiltonian corresponding to one eigenvalue of ˆ
L 2 and ˆ
S 2 can be
labelled by their values for L and S. It has become customary to use as labels not the
spin moment S and orbital moment L but rather the spin multiplicity 2S + 1 and L,
in the notation 2S+1 L, where a spectroscopic notation S, P, D, F, G, ... is used for
L = 0, 1, 2, 3, 4,... The degenerate set of (2S + 1) × (2L + 1) eigenfunctions of
2S+1 L is commonly called an LS term. Examples of free atom (ion) LS terms are 1 S,
2 P, 4 F, ...
Since there is a one-to-one correspondence of the different L eigenvalues with the
irreducible representations (IR) of the spherical symmetry group SO(3), the labels
of the LS terms are simultaneously symmetry labels. Note that the angular moment
operators ˆ
L x , ˆ
L y , ˆ
L z transform as the rotation operators ˆ
R x , ˆ
R y , ˆ
R z , i.e. as P [1]. The
behavior of atoms and ions, free and in compounds, depends for a large part on the
ground term and the lowest excited terms. The symmetry of these lowest LS terms
and the ordering of their energies can in general be well deduced using a simple one
configuration model.
So far we have not considered any relativistic effects for the atoms. In particular, spin appears in the wave function but not in the non-relativistic Hamiltonian.
Traditionally, spin is introduced ad hoc to explain the splitting of a beam of silver
atoms into two parts in the famous experiment of Stern and Gerlach in 1922. A similar splitting was observed for a beam of hydrogen atoms in a later experiment. The
splitting indicates the presence of an angular moment, but it cannot be an orbital
angular moment since both atoms have an L = 0 ground state. Therefore, the spin
property introduced to explain the splitting is considered to reflect an intrinsic angular moment s, which for electrons must be 1/2, and hence it is concluded that each
electron has an additional quantum number s = 1/2, with a z-component s z of either
+1/2or−1/2. The individual spin angular moments of the electrons in an atom can
be coupled together to give a total spin angular moment S, analogous to the coupling
of the individual orbital angular moments l to a total angular moment L. Since there
is no spin operator in the Hamiltonian, there is also no coupling between spin and
orbital angular moment.
35
field, for the time being neglecting any spin-orbit coupling. The total orbital angular
moment operators ˆ
L 2 , ˆ
L z and the total spin angular moment operators ˆ
S 2 and ˆ
S z commute with the Hamilton operator ˆ
H where each electron moves in a field of spherical
symmetry due to the nucleus and the field due the other electrons. Therefore, the
eigenfunctions of ˆ
H are in general also eigenfunctions of the other four operators.
Only in the case of degenerate eigenstates of ˆ
H one may choose (or find) eigenfunctions that are not simultaneously eigenfunctions of the other four operators. In that
case, however, the eigenfunctions of ˆ
H may always be rotated within the degenerate
set to become also eigenfunctions of the other operators.
This implies that these eigenfunctions of ˆ
H can be labelled using the quantum
numbers S, M S , L and M L . The energy eigenvalues only depend on the eigenvalues
of ˆ
L 2 and ˆ
S 2 , and not on M S and M L . Therefore, the degenerate set of eigenfunctions
of the free-atom Hamiltonian corresponding to one eigenvalue of ˆ
L 2 and ˆ
S 2 can be
labelled by their values for L and S. It has become customary to use as labels not the
spin moment S and orbital moment L but rather the spin multiplicity 2S + 1 and L,
in the notation 2S+1 L, where a spectroscopic notation S, P, D, F, G, ... is used for
L = 0, 1, 2, 3, 4,... The degenerate set of (2S + 1) × (2L + 1) eigenfunctions of
2S+1 L is commonly called an LS term. Examples of free atom (ion) LS terms are 1 S,
2 P, 4 F, ...
Since there is a one-to-one correspondence of the different L eigenvalues with the
irreducible representations (IR) of the spherical symmetry group SO(3), the labels
of the LS terms are simultaneously symmetry labels. Note that the angular moment
operators ˆ
L x , ˆ
L y , ˆ
L z transform as the rotation operators ˆ
R x , ˆ
R y , ˆ
R z , i.e. as P [1]. The
behavior of atoms and ions, free and in compounds, depends for a large part on the
ground term and the lowest excited terms. The symmetry of these lowest LS terms
and the ordering of their energies can in general be well deduced using a simple one
configuration model.
So far we have not considered any relativistic effects for the atoms. In particular, spin appears in the wave function but not in the non-relativistic Hamiltonian.
Traditionally, spin is introduced ad hoc to explain the splitting of a beam of silver
atoms into two parts in the famous experiment of Stern and Gerlach in 1922. A similar splitting was observed for a beam of hydrogen atoms in a later experiment. The
splitting indicates the presence of an angular moment, but it cannot be an orbital
angular moment since both atoms have an L = 0 ground state. Therefore, the spin
property introduced to explain the splitting is considered to reflect an intrinsic angular moment s, which for electrons must be 1/2, and hence it is concluded that each
electron has an additional quantum number s = 1/2, with a z-component s z of either
+1/2or−1/2. The individual spin angular moments of the electrons in an atom can
be coupled together to give a total spin angular moment S, analogous to the coupling
of the individual orbital angular moments l to a total angular moment L. Since there
is no spin operator in the Hamiltonian, there is also no coupling between spin and
orbital angular moment.