Russell-Saunders coupling

In atomic spectroscopy, Russell–Saunders coupling, also known as L–S coupling, specifies a coupling scheme of electronic spin- and orbital-angular momenta.

In Russell-Saunders coupling—called after H. N. Russell and F. A. Saunders—the orbital angular momentum eigenstates of the electrons are coupled to eigenstates with quantum number L of the total angular momentum operator squared L². Separately the one-electron spin functions are coupled to eigenstates with quantum number S of total S². Sometimes there is further coupling to J ≡ L + S. The resulting L-S eigenstates are characterized by term symbols.

As an example we consider the excited helium atom in the atomic electron configuration 2p3p. By the triangular conditions the one-electron spins s = ½ can couple to |½−½|,  ½+½ = 0,  1 (spin singlet and triplet) and the two orbital angular momenta l = 1 can couple to L = |1−1|, 1, 1+1 = 0, 1, 2. In total, Russell-Saunders coupling gives two-electron states labeled by the term symbols:

¹S, ¹P, ¹D, ³S, ³P, ³D,

The dimension is 1×(1+3+5) + 3×(1+3+5) = 36. The electronic configuration 2p3p stands for 6×6 = 36 orbital products, as each of the three p-orbitals has two spin functions, so that in total there are 6 spinorbitals with principal quantum number n = 2 and also 6 spinorbitals with n = 3. A check on dimensions before and after coupling is useful because it is easy to overlook coupled states.

Russell-Saunders coupling gives useful first-order states in the case that one-electron spin-orbit coupling is much less important than the Coulomb interactions between the electrons and can be neglected. This occurs for the low Z part (i.e., the lighter elements) of the periodic table, roughly up to the lanthanoids (previously lanthanides, starting at Z = 57) . The usefulness stems from the fact that states of different L and S do not mix under the total Coulomb interaction, so that L-S coupling achieves a considerable block diagonalization of the matrix of a Hamiltonian in which spin-orbit coupling is absent.

In the high Z regions of the periodic system it is common to first couple the one-electron momenta j ≡ l + s and then the one-electron j-eigenstates to total J. This so-called j-j coupling scheme gives a more useful first-order approximation when spin-orbit interaction is larger than the Coulomb interaction and spin-orbit interaction is included, while the Coulomb interaction is neglected. If, however, in either coupling scheme all resulting states are accounted for, i.e., the same subspace of Hilbert (function) space is obtained, then the choice of coupling scheme is irrelevant in calculations where both interactions—electrostatic and spin-orbit—are included on equal footing.

More complicated electron configurations

Spin branching diagram. Number of electrons n on the abscissa. Spin quantum number S on the ordinate.

Spin coupling

Let us consider how to couple n spin-½ particles (electrons) to eigenstates of total S².

The one-electron system (n = 1) has two functions with s = ½ and m = ± ½ (spin up and down). We have seen that addition of an electron leads to four two-electron spin functions: a singlet (2S + 1 = 1) and a triplet space (a "ladder" of 2S + 1 = 3 spin functions). If we add one spin to the two-electron triplet (S = 1), the triangular conditions tell us that S = 3/2, and S = 1/2 can be obtained. In the diagram on the right we see that two lines depart from the n = 2, S = 1 node. One line goes up and joins the n = 3, S = 3/2 node, the other goes down and joins the n = 3, S = 1/2 node.

A two-particle spin system also has a spin-singlet S = 0. Adding a spin to it leads to a doublet, S = 1/2. It can be shown that the doublet thus obtained is orthogonal to the doublet with the intermediate (n = 2, S = 1) spin triplet. Two different paths lead two different—even orthogonal—three-electron spin doublets. This is why the number 2 is listed in the circle at the n = 3, S = 1/2 node.

From the two doublets we can, by again adding one electron, create two four-electron singlets, and two four-electron triplet spaces (ladders). A four-electron triplet space can also be obtained from the three-electron quadruplet (S = 3/2) by subtracting S = 1/2. Hence we see the number 3 at the n = 4, S = 1 node. It is the sum of numbers in the nodes directly on its left connected to it.

The system of sequentially coupling one electron at the time should be clear now. Every path in the branching diagram corresponds to a unique (2S+1)-dimensional ladder of eigenfunctions of total spin angular momentum operator S² with spin quantum number S. Ladders belonging to different paths are orthogonal. The number in the circle indicates how many paths have this node as endpoint.

It is of interest to check dimensions. For instance, the total spin space of five electrons is of dimension 2⁵ = 32. Reading from bottom to top in the diagram, we find 5×2 + 4×4 + 1×6 (second factors being 2S+1), which indeed adds up to 32.

The actual coupling of electron n to a state of n − 1 electrons uses special values of Clebsch-Gordan coefficients. Let us use k as a path index (e.g., for S = 1 and n = 6, the index k runs from 1 to 9). Then addition (line upwards) gives the n-electron spin state:

$|k,S,M\rangle ={\sqrt {\frac {1}{2S}}}{\Big (}{\sqrt {S+M}}\,|k,S-{\tfrac {1}{2}},M-{\tfrac {1}{2}}\rangle |{\tfrac {1}{2}},{\tfrac {1}{2}}\rangle +{\sqrt {S-M}}\,|k,S-{\tfrac {1}{2}},M+{\tfrac {1}{2}}\rangle |{\tfrac {1}{2}},-{\tfrac {1}{2}}\rangle {\Big )}.$

Subtraction (line downwards) gives the n-electron spin state:

${\begin{aligned}|k,S,M\rangle =&{\sqrt {\frac {1}{2S+2}}}{\Big (}-{\sqrt {S-M+1}}\,|k,S+{\tfrac {1}{2}},M-{\tfrac {1}{2}}\rangle |{\tfrac {1}{2}},{\tfrac {1}{2}}\rangle \\&\quad +{\sqrt {S+M+1}}|k,S+{\tfrac {1}{2}},M+{\tfrac {1}{2}}\rangle |{\tfrac {1}{2}},-{\tfrac {1}{2}}\rangle {\Big )}\end{aligned}}$

In particular, for n = 2 we find the spin triplet:

$|1,1,M\rangle ={\sqrt {\frac {1}{2}}}{\Big (}{\sqrt {1+M}}\,|1,{\tfrac {1}{2}},M-{\tfrac {1}{2}}\rangle |{\tfrac {1}{2}},{\tfrac {1}{2}}\rangle +{\sqrt {1-M}}\,|1,{\tfrac {1}{2}},M+{\tfrac {1}{2}}\rangle |{\tfrac {1}{2}},-{\tfrac {1}{2}}\rangle {\Big )},$

which gives the triplet "ladder", where we suppress in the notation the index k = 1,

${\begin{aligned}|1,1\rangle &=|{\tfrac {1}{2}},{\tfrac {1}{2}}\rangle |{\tfrac {1}{2}},{\tfrac {1}{2}}\rangle \equiv \alpha (1)\alpha (2)\\|1,0\rangle &={\sqrt {\tfrac {1}{2}}}{\Big (}|{\tfrac {1}{2}},-{\tfrac {1}{2}}\rangle |{\tfrac {1}{2}},{\tfrac {1}{2}}\rangle +|{\tfrac {1}{2}},{\tfrac {1}{2}}\rangle |{\tfrac {1}{2}},-{\tfrac {1}{2}}\rangle {\Big )}\equiv {\sqrt {\tfrac {1}{2}}}{\Big (}\beta (1)\alpha (2)+\alpha (1)\beta (2){\Big )}\\|1,-1\rangle &=|1,{\tfrac {1}{2}},-{\tfrac {1}{2}}\rangle |{\tfrac {1}{2}},-{\tfrac {1}{2}}\rangle \equiv \beta (1)\beta (2)\end{aligned}}$

Here the more common notation for one-electron spin functions (α for m = ½ and β for m = −½) is introduced. Note that all three triplet spin functions are symmetric under interchange of the two electrons.

The two-electron spin singlet follows from (suppressing the path index k = 1 again):

$|0,0\rangle ={\sqrt {\tfrac {1}{2}}}{\Big (}-|{\tfrac {1}{2}},-{\tfrac {1}{2}}\rangle |{\tfrac {1}{2}},{\tfrac {1}{2}}\rangle +|{\tfrac {1}{2}},{\tfrac {1}{2}}\rangle |{\tfrac {1}{2}},-{\tfrac {1}{2}}\rangle {\Big )}\equiv {\sqrt {\tfrac {1}{2}}}{\Big (}\alpha (1)\beta (2)-\beta (1)\alpha (2){\Big )}$

a function that is antisymmetric (changes sign) under permutation of electron 1 and 2.

(To be continued)

Orbital coupling

The one-electron orbital angular momenta l can also be coupled sequentially, i.e., one after the other. Different paths give orthogonal states. Consider as an example the configuration p³. Repeated application of the triangular conditions gives that 3³ = 27 dimensional space decomposes as follows in eigenspaces of L², [with L ≡ l(1) + l(2)+ l(3) ]:

(2)F,   (2)D,   (1)D,   (2)P,   (1)P,   (0)P,   (1)S

where the two-electron quantum numbers L₁₂ of electron 1 and 2 precede in brackets the letters designating the three-electron angular momentum quantum number. Dimensions are: 7 + 5 + 5 + 3 + 3 + 3 + 1 = 27. We see that there are two orthogonal D spaces and three orthogonal P spaces, while there is only one F and S space (of dimension 7 and 1, respectively).

The algorithm should be clear now. For instance, to decompose the 135-dimensional space belonging to the configuration p³d, one can use the result for p³ and add d to any of the eigenspaces and use the triangular conditions again, as in F × d gives L = 5, 4, 3, 2, 1.