Spin-orbit coupling permutational symmetry

Spin multiplicity, permutational symmetry, dynamic Jahn-Teller and geometric phase effects, 706-711 Spin-orbit coupling conical intersections ... 

The present work details the derivation of a full coupled-cluster model, including single, double, and triple excitation operators. Second quantization and time-independent diagrams are used to facilitate the derivation the treatment of (diagram) degeneracy and permutational symmetry is adapted from time-dependent methods. Implicit formulas are presented in terms of products of one- and two-electron integrals, over (molecular) spin-orbitals and cluster coefficients. Final formulas are obtained that restrict random access requirements to rank 2 modified integrals and sequential access requirements to the rank 3 cluster coefficients. 

Once more, it is important to emphasize that, in computing the GVB wave-function, no restrictions have been imposed other that the four electrons should be spin-coupled in order to produce a total spin equal zero and that each orbital should contain no more than one electron, in order to satisfy the permutation symmetry requirement. 

The Hamiltonian (O Eq. 2.23) maintains full symmetry and is invariant under electronic permutations and under rotation-reflections of the electronic coordinates. Trial functions are usually constructed from atomic orbitals and from their spin-orbitals. Permutational antisymmetry is achieved by forming Slater determinants from the spin-orbitals. Rotational symmetry is usually realized by vector coupling of orbitals that form bases for representations of the rotation group SO(3). Spin-eigenfunctions too are achieved by vector coupling. ... 

We turn now to a brief discussion of the symmetry properties of spin-coupled orbitals . Because they almost always have the form of deformed atomic functions, the effect of a spatial symmetry operation upon the orbitals is to permute them amongst themselves. Thus each symmetry operation corresponds to a certain permutation P. From this it can be seen that the effect of upon complete spin-coupled wavefunctions is to transform them amongst themselves as follows ... 

---