Tensor operator doublet

The condition j + j > 1 for a matrix element of a first rank tensor operator implies, e.g., that there is no first-order SOC of singlet wave functions. Two doublet spin wave functions may interact via SOC, but the selection rule /+ / > 2 for i (2)(Eq. [171]) tells us that electronic spin-spin interaction does not contribute to their fine-structure splitting in first order. 

We conclude that T - lvac) - provided that it does not vanish - represents a tenscH- state with spin eigenvalues 5 and M. Because of the close relationship between spin tensw operators and spin eigenfunctions, the terminology for spin functions is often used for spin tensor operators as well. Thus, a spin tensor operator with 5 = 0 is referred to as a singlet operator, S — gives a doublet operator, 5 = 1 a triplet, and so on. 

Comparing these expressions with (2.3.1) and (2.3.2), we eonclude that the creation operators a, constitute a doublet tensor operator. Applied to the vacuum state, these operators generate one-electron doublet states. In accordance with the general relation (2.3.11), we obtain a new doublet by conjugation and reordering of ] ,. p - We find that the pair —flp, Opa) constitutes a doublet tensor ... 

For strings containing two or more elementary operators, it is possible to constmct more than one tensor operator. We shall in Section 2.6.7 present a general method for the construction of tensor operators from strings of elementary operators. At present, we note that, by coupling the two doublet operators and it is possible to generate a singlet two-body creation... 

Hermitian and unchanged by the operation. The notation used here conforms with the finite group irreducible tensor scheme 38, 39). We have already proved that cannot split the doublets and need therefore only discuss Now the states concerned are... 

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