## Symmetry of Atomic States

Atomic States have symbols such as (2S+1)LMs.   Thus a vanadium atom, 3d34s2 would give rise to the states: 4P, 4D, 4F, 4G, 4H, 6S, and 6D.  Some of these States occur more than once, for example the 4P State occurs four times. All components of a single State that have the same spin, S, and angular momentum, L, are degenerate. Following Griffith, "The Theory of Transition Metal Ions", the L orbital angular momenta values are represented in MOPAC by the letters S, P, D, F, G, H, I, K, L, M corresponding to the angular momenta L = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.  Note that symbol "J" is missing, this is because it is used extensively in other aspects of atomic theory.

All atomic States that can exist for a s-p-d basis set with any allowed number of electrons are automatically identified. By specifying the component of spin, using Ms=n.n, individual magnetic components within a manifold such as 6S can be identified.

Atomic theory is both beautiful and complicated.  Simple spin-spin and orbit-orbit coupling, i.e., Russell-Saunders or L-S coupling, can be modeled, but relativistic coupling (j - j coupling) cannot be modeled, due the the absence of the spin-orbit interaction terms.  Despite the lack of spin-orbit interactions, a lot of elegant symmetry work can still be done.  One of the most beautiful sections of atomic symmetry theory involves the coupling of two space angular momenta  Coupling coefficients involve the Wigner or Clebsch-Gordan 3j symbol <L1L2m1m2|L1L2Lm> which equals:

{(L+m)!(L-m)!(L1-m1)!(L2-m2)!(L1+L2-L)!(2L+1)/((L1+m1)!(L2+m2)!(L1-L2+L)!(L2-L1+L)!(L1+L2+L+1)!)}0.5
×Δ(m,m1+m2)Σr(-1)(L1+r-m1){((L1+m1+r)!(L2+L-r-m1)!)/(r!(L-m-r)!(L1-m1-r)!(L2-L+m1+r)!)}

with 0! = 1.  All aspects of this can be modeled.  For users of MOPAC who are brave enough to explore this topic, two tools are provided:

A set of Wigner coefficients otherwise known as the Clebsch-Gordan or Wigner 3j coefficients for all values of l1 and l2 up to 6.  This is a large file, over 1Mb.  Each coefficient is expressed four ways:

(A) As a decimal.
(B) As a rational fraction.
(C) As a rational fraction with a common denominator
(D) As a rational fraction expressed as powers of prime numbers, thus the coefficient for the most complicated term, <6,4,5,-3|6,5,9,1>, is (367^2)/(2^2.3.5.7.11.13.17)

A FORTRAN 90 source code to generate these coefficients.

An illustrative exercise in atomic theory is to use the shift step-down operators, I-,  to move through the associated Hilbert spin-space, e.g., to go from 6S with Ms = 5/2 to Ms = -5/2, then show that the State is annihilated on application of the step-down operator once more.  Moving through the orbital angular momentum space is tedious, and definitely not worth doing.