Spin-Orbit Coupling and Term Symbols
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The interaction between the spin angular momentum quantum numbers of two electrons gives rise to states of so called different spin multiplicity, such as singlet and doublet states.

The spin angular momentum of an electron may also interact with the orbital angular momentum of the electron to split the energies of the different levels, causing different transition energies to be observed. This effect is known as spin-orbit coupling.

The interaction of the spin angular momentum and the orbital angular momentum gives rise to the total angular momentum: when the spin and orbital angular momentum are parallel, the total angular momentum is high.

For a given electron the total angular momentum, j, is either l+s or l-s (with s = 1/2 for the electron), and the energy level corresponding l is therefore split into two terms for the different values of j.

The energy of the state characterized by the quantum numbers, l, s and j, is given by the expression:

Here, A is the spin-orbit coupling constant, and the degree of coupling increases as Z4 (where Z is the atomic number).

The total angular momentum of a single electron can also couple with the total angular momentum of other electrons, such that the total angular momentum of an atom as a whole can have a range of values, depending upon the exact occupation of the orbitals. The spin, orbital, and total angular momentum for an atom or ion in a given electronic configuration is expressed as a term symbol.

In light atoms, the spin-orbit coupling in individual electrons is weak, and instead the coupling of the spin angular momenta of different electrons, or the coupling of the orbital angular momenta of different atoms, is stronger. This type of spin-orbit coupling is known as the Russell-Saunders scheme, and gives rise to term symbols as shown, for a species with two electrons in d orbitals, in the table below.

Russell Saunders coupling: the d2 ion

When more than one valence electron is present, interactions between the electrons result in couplings between the quantum numbers for the individual electrons. The quantum state of the overall ion depends on the quantum states of the individual electrons.

The quantum state of the electron is determined by the values of n (the principal quantum number), l (the orbital angular momentum quantum number), ml (the magnetic quantum number), and s (the spin quantum number).

There may be coupling between the spin angular momenta of two electrons, spin-spin coupling, the orbital angular momenta of two electrons, orbit-orbit coupling, and the spin and orbital angular momenta of the same electron, spin-orbit coupling.

In the Russell-Saunders scheme, the case for the first row transition elements, and in general for elements up to atomic number 30, the magnitude of coupling is assumed to be in the order:

spin-spin coupling > orbit-orbit coupling > spin-orbit coupling

Spin-spin coupling:

The spin quantum number, S, for a system of electrons is calculated from the spin quantum numbers, s1 and s2, for the separate electrons according to

for the d2 system,

Orbit-orbit coupling:

For two electrons with orbital angular momentum quantum numbers l1 and l2, the total orbital angular momentum quantum number, L, is

This is known as the Clebsh-Gordan Series.

for the d2 system,

l1 = l2 =2, so

Different values of L are referred to by different term letters: S (L=0), P (L=1), D (L=2), F (L=3), G (L=4), H (L=5), ...

the d2 system has G, F, D, P, and S states

Spin-orbit coupling:

The total angular momentum quantum number, J, is obtained by coupling the total spin and orbital angular momenta according to:

Different values of S can have different numbers of values of J, or different numbers of levels. The number of levels possible for a given S number is the multiplicity, given by (2S + 1).

the d2 system has multiplicity values

S = 1: (2S+1) = 3 (a triplet)

S = 0: (2S+1) = 1 (a singlet)

The information on the possible values of S, L and J as summarized in the term symbol:

Not all terms are allowed, as some would require electrons with the same spin to occupy the same orbital, in contravention of the Pauli exclusion principle.

the d2 system has the possible terms:

3P, 3F, 1S, 1D, 1G

The relative order of the energies of these terms is given by Hund's rules:

1) The most stable state is the one with the maximum multiplicity

2) For a group of terms with the same multiplicity, the one with the largest value of L lies lowest in energy.

the ground state term for the d2 system is:

3F

An important consequence of term symbols is their use to express the ranges of S, L, and J which may be involves in allowed transitions between the levels the term symbols represent. These allowed transitions may be summarized as a set of selection rules:

ΔS = 0

ΔL = 0, +1, -1 with Δl = +1, -1

ΔJ = 0, +1, -1 but J = 0 to J = 0 is forbidden

These selection rules only apply in the Russell-Saunders coupling scheme. In heavier atoms, the coupling between the spin and orbital angular momentum of individual electrons is much stronger, and only the total angular momentum, J, is important. The selection rules based on the values of S and L therefore do not hold. A better coupling scheme for the heavy atoms is jj-coupling, where the total angular momentum of each electron is calculated first, and then these are coupled to give the overall total angular momentum of the atom.

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