General structure of perturbation operators

We shall assume that the perturbation operators appearing in (20) are all one-electron operators. In second quantization they can therefore be written as 

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 

To signal the transition from a summation over individual orbitals to a summation over Kramers pairs I will employ capital letters, but only under the summation sign Y,pq Y,pq I retain lowercase orbital indices for both cases, as it poses no confusion. We may further insist that the perturbation operators hx have a specific symmetry with respect to time reversal 

One way of explicitly building time reversal symmetry into the formalism is to introduce Kramers replacement operators [83,81] in analogy with the singlet and triplet excitation operators of the non-relativistic domain [80]. Using (181), we may rewrite the property (179) operator as 

The formalism can be straightforwardly extended to two-electron operators with the introduction of Kramers double replacement operators [81]. However, the multitude of terms arising in subsequent derivations finally leads to a rather cumbersome formalism. In the author s opinion it is better to derive general formulas and then consider the structure due to time reversal symmetry after the derivation. 

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