Triplet operators

The relations of Eq. (5.17) allow the scaling of all components of a tensor operator with the same factor. The factor used above is the one commonly used. It is noted that if i and j are identical the triplet operator of Eq. (5.24) vanishes. 

The arbitrary scaling factors on the above tensor operators are again chosen in accordance with the general use. It is seen that the triplet operator is defined also for the case where i and j are identical. The singlet operator, Sjj(0,0) is proportional to the operator E j, obtained from considering spin... 

Although Eq. (25) looks like its nonrelativistic counterpart, the A and B matrices are complex and their elements have to be calculated using DHF). The K operator cannot be expressed in terms of the usual singlet and triplet operators and, as the Hamiltonian does not commute with the spin operator, there is only one type of A and B matrices. 

It has been shown, using the Wigner-Eckart theorem, that (a) is identically zero, and (e) could be expected to be negligible owing to the presence of the SD operator twice. The (c) contribution, which involves a combination of three triplet operators, was not calculated since it is expected to be smaller than (b) because it also contains the SD operator. Thus, only the (b) and (d) terms were calculated to estimate the SO correction. Furthermore, an important simplification was obtained taking the effect of the two-electron part of the SO interaction as a shielding of the nucleus field, which reduces the one-electron part. To this end, an effective charge for the N nucleus was introduced, Eq. (32). 

It follows from the symmetry properties of the 3 symbols and from the anticommutation relations of the electron field operators that the triplet operators do not exist for L even, while the singlet operators exist only for such L. 

The symmetry of the spin-orbit operator was derived in section 10.3. The spatial part of this operator transforms as the vector of rotations R = (Rx, Ry, Rz)- This means that the spin-orbit operator will connect states of different spatial symmetry. In C2v, for example, states of all spatial symmetries are connected by the spin-orbit operator. In D2h, the gerade states are all connected by the spin-orbit operator, and likewise the ungerade states, but there is no connection between gerade and ungerade states because the spin-orbit operator commutes with the inversion operator (it is an even operator). The spin operator transforms as a spherical tensor of rank 1 it is essentially a triplet operator. Therefore, it can connect states whose S and Ms values differ by 0 or 1. 

The singlet two-body creation operator (2.3.16) is symmetric in the indices p and q whereas the triplet operators are antisymmetric. For p =, the singlet two-body creation operator becomes... 

From the orbital density matrices considered in Section 2.7.1, we may calculate expectation values of singlet operators. For triplet operators such as the Fermi contact operator, a different set of density matrices is needed. Consider the evaluation of the expectation value for a one-electron triplet operator of the general form... 

Show that the three components of the triplet operator become ... 

Construct the two-electron singlet operator (i, 1,, 0) and express it in terms of the one-electron triplet operators ... 

Alternatively, we may write k in terms of the Cartesian triplet operators (2.3.27). Since the Cartesian components obey the simple conjugation relation (2.3.28), we now obtain the following more symmetric form for a general anti-Heimitian operator k ... 

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