Singlet operators

A tensor operator with S=0 is called a singlet operator, an operator with S... 

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... 

In the non-relativistic domain one-electron operators can be classified as triplet and singlet operators, depending on whether they contain spin operators or not. In the relativistic domain the spin-orbit interaction leads to an intimate coupling of the spin and spatial degrees of freedom, and spin symmetry is therefore lost. It can to some extent be replaced by time-reversal symmetry. We may choose the orbital basis generating the matrix of Hx to be a Kramers paired basis, that is each orbital j/p comes with the Kramers partner = generated by the action of the time-reversal operator We can then replace the summation over individual orbitals in (178) by a summation over Kramers pairs which leads to the form... 

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. 

These Ep operators are the equivalent of the singlet and triplet excitation operators of nonrelativistic theory is the singlet operator and the remaining three are the triplet... 

Two features of this representation of E2 are noteworthy. First, we still have the same number of terms as in the original primitive representation. Second, we have a mixture of and E q operators, and thus we have not achieved a basis set reduction, while the nonrelativistic Epq allowed us to formally halve the size of the basis. The conclusion is that even if it is possible to construct operators analogous to Epq for the relativistic case, there is less, if anything, to be gained compared with the nonrelativistic case because we must evaluate expressions involving both triplet and singlet operators. This is to be expected because the operators include spin-dependent terms. These excitation operators have their uses in correlated methods, and there are reductions that occur due to point group symmetry, which will be considered later. 

Since the Hamiltonian H is a number-conserving singlet operator, it commutes with both N and Sx and therefore with Uap- Equation (11.8.69) may then be written as... 

To ensure that the coupled-cluster state has the same symmetry as the reference state for any choice of amplitudes, we must require the excitation operators in the cluster operator to he singlet operators, satisfying (2.3.1) and (2.3.2) ... 

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. 

Since this Hamiltonian is a singlet operator (see Section 2.3.4), it commutes with the operators for the total and projected spins ... 

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... 

Construct the two-electron singlet operator (5- 5 O) and show that it may be expressed... 

Use the properties of spin tensor operators to verify the following relationship between the matrix elements of a singlet operator... 

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