Cartesian triplet spin functions

The three multiplet components of an excited triplet state are degenerate in zeroth order. We have therefore, in principle, the freedom of choosing these in their spherical or Cartesian forms. On the other hand, the spin-orbit split triplet levels will transform according to the irreps of the molecular point group. For a smooth variation of the wave function gradient with respect to the perturbation parameter X, we employ Cartesian triplet spin functions also in the unperturbed case and express them as ket vectors ... 

A look at Table 6 tells us that the Ms = + 1 (aa) and Ms = -1 (pp) components of the triplet spin function do not separately transform according to one of the irreps of C2v. Their positive linear combination, aa + pp, exhibits f>i character, the negative one, aa-pp, belongs to the B2 irrep. These linear combinations closely resemble expressions [37] for Cartesian tensor components. In accordance with the tensor normalization and phase conventions, we choose Ty+ (aa+pp) and Tx = j(PP — aa). Among the M = 0 two-electron functions, the singlet function So = (ap — pa) behaves totally symmetrically (Aj), whereas the Ms = 0 component of the triplet, To = Tz = (ap+ pa), transforms according to A2. 

The function F(l,2) is in fact the space part of the total wave function, since a non-relativistic two-electron wave function can always be represented by a product of the spin and space parts, both having opposite symmetries with respect to the electrons permutations. Thus, one may skip the spin function and use only the space part of the wave function. The only trace that spin leaves is the definite per-mutational symmetry and sign in Eq.(14) refers to singlet as "+" and to triplet as Xi and yi denote cartesian coordinates of the ith electron. A is commonly known angular projection quantum number and A is equal to 0, 1, and 2 for L, II and A symmetry of the electronic state respectively. The linear variational coefficients c, are found by solving the secular equations. The basis functions i(l,2) which possess 2 symmetry are expressed in elliptic coordinates as ... 

---